text
stringlengths 6
976k
| token_count
float64 677
677
| cluster_id
int64 1
1
|
|---|---|---|
About Concrete Mathematics: A Foundation for Computer Science
The primary aim of this book's well-known authors is to provide a solid and relevant base of mathematical skills -- the skills needed to solve complex problems, to evaluate horrendous-looking sums, to solve complex recurrence relations, and to discover subtle patterns in data.
About Concrete Mathematics: A Foundation for Computer Science | 677.169 | 1 |
Learning Support Mathematics
The Learning Support Mathematics program assists students in developing the ability to perform mathematical computations, use measurements, make estimates and approximations, judge reasonableness of results, formulate and solve mathematical problems, select appropriate approaches and problem-solving tools and use elementary concepts of probability and statistics. Learning Support mathematics courses are intended for students who need additional preparation in mathematics prior to enrolling in college-level courses.
Learning Support Mathematics Course Descriptions
Courses below constitute the TBR required sequence based on college placement. | 677.169 | 1 |
Mathematics For Elementary Teachers - 05 edition
Summary: The goal of this text is to provide prospective elementary teachers with a deep understanding of the mathematics they will be called on to teach. Through a careful, mathematically precise development of concepts, this text asks that students go beyond simply knowing how to carry out mathematical procedures. Students must also be able to explain why mathematics works the way it does. Being able to explain why is a vital skill for teachers. Through activities, examples...show more and applications, the author expects students to write and solve problems, make sense of the mathematics, and write clear, logical explanations of the mathematical concepts. The accompanying Activities Manual promotes engagement, exploration, and discussion of the material, rather than passive absorption. Both students and instructors should find this material fun, interesting, and rewarding | 677.169 | 1 |
Additional product details
Fundamentals of Precalculus is designed to review the fundamental topics that are necessary for success in calculus. Containing only five chapters, this text contains the rigor essential for building a strong foundation of mathematical skills and concepts, and at the same time supports students' mathematical needs with a number of tools newly developed for this revision. A student who is well acquainted with the material in this text will have the necessary skills, understanding, and insights required to succeed in calculus. CourseSmart textbooks do not include any media or print supplements that come packaged with the bound book. | 677.169 | 1 |
Elementary Linear Algebra Applications Version
9780471669593
ISBN:
0471669598
Edition: 9 Pub Date: 2005 Publisher: John Wiley & Sons Inc
Summary: This classic treatment of linear algebra presents the fundamentals in the clearest possible way, examining basic ideas by means of computational examples and geometrical interpretation. It proceeds from familiar concepts to the unfamiliar, from the concrete to the abstract. Readers consistently praise this outstanding text for its expository style and clarity of presentation. The applications version features a wide ...variety of interesting, contemporary applications. Clear, accessible, step-by-step explanations make the material crystal clear. Established the intricate thread of relationships between systems of equations, matrices, determinants, vectors, linear transformations and eigenvalues | 677.169 | 1 |
Lesson Plans & Activities
TAKS 9th Grade Math Objective 1 -10
Objective 1: The student will describe functional relationships in a variety of ways.
Objective 2: The student will demonstrate an understanding of the properties and attributes of functions.
Objective 3: The student will demonstrate an understanding of linear functions.
Objective 4: The student will formulate and use linear equations and inequalities.
Objective 5: The student will demonstrate an understanding of quadratic and other nonlinear functions.
Objective 6: The student will demonstrate an understanding of geometric relationships and spatial reasoning.
Objective 7: The student will demonstrate an understanding of two- or three- dimensional representations of geometric relationships and shapes.
Objective 8: The student will demonstrate and understanding of the concepts and uses of measurement and similarity.
Objective 9: The student will demonstrate an understanding of percents, proportional relationships, probability, and statistics in application problems.
Objective 10: The student will demonstrate an understanding of the mathematical processes and tools used in problem solving. | 677.169 | 1 |
Description The first half of a modern high school algebra sequence with a focus in seven major topics: transition from arithmetic to algebra, solving equations & inequalities, probability and statistics, proportional reasoning, linear equations and functions, systems of linear equations and inequalities, and operations on polynomials. Students enrolled in this course must take the WA State High School End of Course Algebra Assessment if they have not attempted it once already. Prerequisite: Must be working toward a high school diploma. | 677.169 | 1 |
User ratings
Review: Euclid's Elements
This is going to be a long term project to get through this. We'll see if I come out on the other side.Read full review
Review: Euclid's Elements
User Review - Rlotz - Goodreads
Euclid's Elements is one of the oldest surviving works of mathematics, and the very oldest that uses axiomatic deductive treatment. As such, it is a landmark in the history of Western thought, and has ...Read full review | 677.169 | 1 |
Chapter 10 Factorisation Techniques
Knowledge of factorisation enables us to learn advanced
mathematics and solve problems which occur in science, business,
computer programming and
engineering. In this chapter, we will consider the highest common
factor, the difference of two squares, factors of quadratic trinomials
over Q, use of perfect squares, factorisation of four terms and
factors of quadratic trinomials over R. | 677.169 | 1 |
Here's a real-world lesson using a business simulation. Two business accounts are used to find slope and intercept functions. The class graphs and interprets the information to find a break even point. There are plenty of worksheets and assessments included in this lesson.
In this recognizing idioms worksheet, students match beginning and ending phrases and sentences by drawing lines to connect each word or words in the left column to words in the right columns. Students create twenty-two idioms. In addition, students may choose from three functional activity ideas.
Design an experiment to model a leaky faucet and determine the amount of water wasted due to the leak. Middle schoolers graph and write an equation for a line of best fit. They use their derived equation to make predictions about the amount of water that whould be wasted from one leak over a long period of time or the amount wasted by serveral leaks during a specific time period.
In this math worksheet, students give examples of functions that will satisfy given conditions. Students tell the tabulations for a given function. Students use the definition of a derivative to compute the inverse of a function. they state the Fundamental Theorem of Calculus and describe its usefulness. Students give an example of a function that can not be integrable.
Investigate non-linear functions based upon the characteristics of the function or the representation of the function. The functions are displayed in multiple formats including as graphs, symbols, words, and tables. Learners use written reflection scored on a rubric to assess understanding.
This Mean Value Theorem and Rolle's Theorem worksheet is very thorough in explaining the two Theorums and showing the formulas. There are six prractice problems for classwork, and eight additional problems for homework.
In this continuity instructional activity, students solve 5 short answer problems about continuity. Students determine where functions are continuous and find the limits of functions at points of discontinuity.
Wow! Students examine a geometric sequence. In this geometry lesson, students measure the lengths of strings on a musical instrument and explore the geometric sequence the frets generate. Students compare and contrast geometric sequences and exponential functions. | 677.169 | 1 |
In districts nationwide, as many as 50% of students fail Algebra I the first time and must repeat it—some more than once. What happens to those who are one or more grade levels behind before they begin Algebra I? Intensified Algebra I is a comprehensive program for an extended-time Algebra class that helps students who are significantly behind become successful in algebra within one academic year. It transforms the teaching of algebra to students who struggle in mathematics.
This program is designed for lower level math students, which I am teaching in inner city Chicago. The curriculum is perfect for my students. The reading is manageable, and the focus is on getting students to really think through problems. | 677.169 | 1 |
The CCGPS Coordinate Algebra Program is a complete set of teacher and student materials developed around the Common Core Georgia Performance Standards (CCGPS) and the Coordinate Algebra Frameworks, Curriculum Map, and Teacher's Guide. The course design has benefited from direct input from Georgia teachers.
Topics are built around accessible core curricula, ensuring that Coordinate Algebra is useful for striving students and diverse classrooms. This program realizes the benefits of exploratory and investigative learning and employs a variety of instructional models to meet the learning needs of students with a range of abilities. Review and purchase
Coordinate Algebra Station Activities for CCGPS The Coordinate Algebra Station Activities for CCGPS is a collection of hands-on, problem-solving activities to provide students with opportunities to practice and apply the math skills and concepts they are learning in their Coordinate Algebra class.
You may use these activities to complement your regular lessons or in place of your regular lessons, if students have the basic concepts but need practice.
The CCGPS Analytic Geometry Program has been organized to coordinate with the CCGPS Analytic
Geometry Frameworks and Curriculum Map. Each lesson includes activities that offer opportunities for exploration and investigation. These activities incorporate concept and skill development and guided practice, then move on to the application of new skills and concepts in problem-solving situations. Throughout the lessons and activities, problems are contextualized to enhance rigor and relevance.
Download the Content Map for an overview of the course units, lessons, sub-lessons, and standards.
CCGPS Advanced Algebra
A complete CCGPS Advanced Algebra Program will be available for school year 2014-2015 and we're working to have Units 1-3 by the end of 2013 for those teaching Accelerated Analytic Geometry. In addition to complete course materials, Digital Enhancements and Online Assessments will also be available.
"I am truly enjoying teaching from the Walch Coordinate Algebra text. I have never found a textbook that matches the Georgia state standards so well. It seems to be tailored to fit our curriculum. The Walch materials are also easily differentiable. This book is usable from accelerated to support classrooms."
- Laura Blair, Math Teacher, Southeast Bulloch High School
"Walch has created the only textbook that truly follows the standards. I have looked at every book available to me and they all simply rearranged the old texts to match up, but did not add or remove anything to ensure that all standards are met. Using the Coordinate Algebra text has made my job so much easier! The chapters are organized in the same order as the CCGPS for Georgia. The Teacher's Edition even provides a timeline so that I know when it is appropriate to pull in the state's tasks. The electronic versions of the student edition and teacher's edition have been a huge help, as well!"
- Heather Lloyd, Math Teacher, Hart County
We are pleased to offer additional high school math resources to Georgia, listed below:
DeKalb County High utilizes Walch High School Science Programs for the GHSGT, covering the Characteristics of Science and the five content domains: Cells and Heredity; Ecology; Structure and Properties of Matter; Energy Transformations; and Forces, Waves, and Electricity.
Summer School and Intercessions
Fulton County uses summer-school support programs for mathematics in grades 6, 7, and 8.
In addition to our GPS-based materials for secondary classrooms, we have a wide variety of research-based materials aligned to national standards that can be readily incorporated into your educational programs. See the results: Clarke County Achieves 12.8% CRCT Gains in Five Weeks... | 677.169 | 1 |
Description: Algebra from the viewpoint of the elementary curriculum with emphasis on proportional and linear relationships. Also included: topic from probability and statistics with connections to other topics in the elementary curriculum. Problem solving is emphasized throughout. | 677.169 | 1 |
Perrine, FL Calculus notations from discrete mathematics are useful in studying and describing objects and problems in branches of computer science, such as computer algorithms, programming languages, cryptography, automated theorem proving, and software development. Conversely, computer implementations... | 677.169 | 1 |
A Basic Course in Statistics 5e
This new edition includes computing exercises at the end of each chapter to reflect the growing use of computers in teaching statistics. It is designed for students taking introductory courses in statistics in school, technical colleges and universities.
For courses in First-year Russian - Introductory Russian. Golosa is a two-volume, introductory Russian-language program that strikes a balance between communication and structure. It is designed to ... | 677.169 | 1 |
The books gdunne have linked to are really great, i also self-studied math using those books. | 677.169 | 1 |
This webinar offers a quick and easy way to learn some of the fundamental concepts for using Maple. Learn the basic steps on how to compose, plot and solve various types of mathematical problems. This webinar will also demonstrate how to create professional looking documents using Maple, as well as the basic steps for using Maple packages.
The typical multivariate calculus course begins with a unit on vectors, lines and planes, material that serves as an introduction to the linear geometry of R³. This webinar will provide solutions for a number of typical, and not-so-typical problems in the "lines-and-planes" section of the calculus course. The problems will be solved with a suite of commands in the Student MultivariateCalculus package, commands specifically designed to handle manipulations of lines and planes in R³. In addition, these problems will also be solved in a syntax-free way via the Context Menu system because the relevant commands have been completely incorporated into that environment.
In this webinar, an introduction is made to modeling, simulation, and analysis with MapleSim by studying a full vehicle model equipped with electric power steering. The webinar will cover modeling and simulation aspects such as a 3D multibody steering mechanism, and multi-domain modeling by inclusion of electrical and 1D translational components.
This webinar, presented by Dr. Robert Lopez, Maple Fellow and Emeritus Professor from the Rose-Hulman Institute of Technology, will provide you with tips and techniques that will help you get started with Maple 17.
The Möbius Project is a revolutionary initiative that brings the power of Maple to even more people, in even more ways. This webinar will demonstrate: how to create Möbius Apps in Maple, how to share Möbius Apps with your colleagues and students using the MapleCloud, and how to grade Möbius Apps in Maple T.A. | 677.169 | 1 |
Mathematical modelling is an essential tool for science. Our researchers collaborate with partners from academia and industry to help tackle real-life problems.
Dr Robert Whittaker, Lecturer
Mathematics is critical to almost every field of human endeavour. Mathematics students at UEA are highly adaptable and high-achieving problem solvers.
Dr Paul Hammerton, Senior Lecturer
The mathematics department is incredibly supportive, offering assistance through problem classes and seminars that really help with coursework.
Kathryn O'Reilly, BSc Mathematics Student
The School of Mathematics at UEA is a flourishing department committed to excellence in teaching and research.
Since the National Student Survey started in 2005, we have consistently featured in the top six mathematics departments in the country. In the most recent National Student Survey, the School achieved an overall student satisfaction of 100% for BSc Mathematics. The School has a strong international reputation for its research and students are taught by leading experts in a broad range of topics in Mathematics. The 2008 Research Assessment Exercise (RAE) judged over half of the research activity in the School to be world-leading or internationally excellent. | 677.169 | 1 |
Mathematics can be viewed as a language for describing the world around us. Indeed, this is largely how mathematics developed. For instance, Calculus was invented by Newton in order to describe how a cannon ball falls to the ground or to describe how the moon orbits the Earth.
This course will be very much in this tradition. We will consider problems or objects that we might observe or encounter every day, for instance: "Why (in terms of the reproductive function of a pine cone) is a pine cone shaped as it is?" Or "Can California water shortages be alleviated by towing icebergs from Antarctica?" Such systems as the human body, the stock market, and sports games are amenable to description, called models, via the mathematics that we encounter early in our college years (and of course, more advanced mathematics can provide more detailed models!).
The goal of this course will be to increase the mathematical literacy of the students taking it. We will provide a set of tools and frameworks with which students can use familiar mathematics to predict and analyze real world problems. The mathematics required will be a "just in time production:" that is, it will be taught when it is needed.
The principal text for this course is Towing Icebergs, Falling Dominoes, and Other Adventures in Applied Mathematics by R.B. Banks. On occasion we will use Slicing Pizzas, Racing Turtles, and Further Adventures in Applied Mathematics also by R.B. Banks, as well as Topics in Mathematical Modeling by K.K. Tung.
Each class will feature a focus problem or focus problems for which we will develop a mathematical model that attempts to describe and predict the system in question. These in-class projects will typically be tackled in teams, and thus attendance for each class is required. In addition to these in-class projects, there will be at least two large modeling problems given to teams for which 5 to 6 page research reports will be required. Some homework- as preparation for the coming discussion- will be required weekly. | 677.169 | 1 |
Math 4xx/6xx
Math 410/610 Complex Analysis (Johnson or Kumjian)
Complex analysis is the study of calculus on the complex numbers. While the theory superficially resembles ordinary calculus of functions on the real numbers, the details and applications are very different. The theory is much richer and more elegant than ordinary calculus, and there is some beautiful geometry involved which has unexpected applications to problems in physics and engineering. Ideas and techniques from complex analysis are used throughout pure and applied mathematics, and in many scientific fields. The familiar pictures from fractal geometry are made using complex analysis. Prerequisites: MATH 283 required; MATH 310 recommended.
Math 430/630 Linear Algebra II (Jabuka or Olson)
The linear algebra behind the following applications will be developed:
1. Network design.
2. Least squares approximations for statistical analysis.
3. The Fast Fourier Transform.
4. Eigenvalues, Eigenvectors, Diagonalization and Jordan Cannonical Forms for matrix exponentiation (differential equations) and powers of matrices (for finite difference
equations).
5. Finite Element Methods.
6. Minimization/Maximization problems.
7. Machine efficient methods for finding eigenvectors and solving the matrix equation Ax = b. [This includes knowing when the coefficient matrix is ill conditioned so that numerical results are likely to be of questionable accuracy, as well as being able to take advantage of certain features of A that occur in various applied problems to improve accuracy and efficiency].
This course will be highly motivated by applications and examples. Prerequisite: MATH 330.
Math 440/640 Topology (Deaconu, Jabuka, Naik)
A topological space is a collection of points and a structure that endows them with some coherence, in the sense that we may speak of nearby points or points that in some sense are close together. We will start by recalling some general notions about sets and functions. Then we will discuss metric spaces, which are natural generalizations of the real line with the usual distance between two points. Finally, we will introduce the general concept of a topological space, studying the notions of continuity, connectedness and compactness. We will have plenty of examples, including familiar curves and surfaces. Prerequisite: MATH 310.
Math 441/641 Algebraic Topology (Jabuka or Keppelmann)
Topology is popularly known as "the rubber sheet geometry". Topologists are jokingly referred to as mathematicians who can't distinguish between a coffee cup and a doughnut, since the surface of one can be continuously stretched ijnto the surface of the other. A doughnut and a two-holed doughnut cannot be stretched into one another; the same goes for a once-punctured disc and a twice-punctured disc. Perhaps surprisingly, group theory (see MATH 331) proves to be a useful tool in the study of questions like this. In this course we will learn two ways of associating algebraic objects to a topological space, namely homology and homotopy. We will study fundamental groups, covering spaces, and homology theory. [SN 2003] Prerequisite: MATH 440/640. Co-requisite: MATH 331.
Math 442/642 Differential Geometry (Herald or Jabuka)
The course will be of interest to: MATH students who want to use their calculus and linear algebra skills to study fascinating geometric problems; COMPUTER SCIENCE students who are into computer graphics and want to come to a deeper understanding of the shapes of curves and surfaces in space; for PHYSICS students who want an introduction to the math behind much of modern theoretical physics. Differential Geometry is used in such diverse fields as Relativity Theory, Mechanics, Control Theory, Computer Graphics, much of Theoretical Physics, as well as in other branches of Mathematics. Prerequisites: MATH 311 or the consent of the instructor. [CH 2001]
Math 449/649 Geometry and Topology -- Knot Theory (Herald, Jabuka or Naik)
We will study the mathematical theory of knots and links and discuss the applications to biology, chemistry and physics. This course will not have any prerequisites, however some background in 300 level math courses (eg. MATH 310, MATH 330) will be necessary. Prerequisites: Consent of the instructor. A corequisite of MATH 311 or 441/641 is strongly recommended. Familiarity with elementary group theory will be useful, but not required. [SN 2001]
Math 461/661 Probability Theory (Formerly Math 451/651) (Kozubowski)
Probability models are used to represent river flows, customer arrivals, atoms, chemical reactions, epidemics, election results, and stock markets. Probability theory is the basis for all of these mathematical models. An infinite series represents the average number of customers arriving this month. An integral gives the average jump in daily stock market prices. A double integral gives the probability that the river will rise higher this year than last year. In this course you will learn to apply the tools of calculus and analysis to problems in probability. Topics to be discussed should include: probability space axioms; random variables; expectation; univariate and multivariate distribution theory; sequences of random variables; Tchebychev inequality; the law of large numbers and the central limit theorem. MATH 461/661 is a beautiful chapter of pure mathematics, as well as the foundation for stochastic processes, statistics, econometrics, and quantum mechanics. Take this course and prepare for a brilliant future in an uncertain world. [TK 2005] Prerequisite: MATH 283. Note: MATH 461 is required for BA and BS in Mathematics (all options) while MATH 661 is required for MS in Mathematics, Statistics option.
Math 466/666 Numerical Methods I (Formerly Math 483/683) (Mortensen, Olson, Telyakovskiy)
This course is a one semester introduction to the subject of Numerical Analysis. Numerical analysis concerns algorithms and methods for obtaining solutions to mathematical problems. We will survey many of the tools and techniques of the field. Some of the topics include interpolation, integration, linear systems, differences, differential equations, nonlinear equations and optimization. This course will be a "hands-on" course focusing on use of the computer, with much less emphasis on theory (an introduction to FORTRAN will be provided). The student will be able to leave this course able to obtain solutions to seemingly intractable problems and understand the basis of many large software packages now available. Prerequisite: MATH 330.
Math 467/ 667 Numerical Methods II (Formerly Math 484/684) (Mortensen, Olson, Telyakovskiy)
Numerical solution of ordinary differential equations; boundary value problems; finite difference methods for partial differential equations; finite element method. Since MATH 466/666 is NOT a prerequisite for this course, there will be a brief review of relevant topics from MATH 466/666, including numerical differentiation and integration. Prerequisite: MATH 285 or equivalent, and a knowledge of computer programming. Note: An introduction to Maple and Matlab will be provided. [JM 2003]
Math 474/674 Sets and Numbers (Deaconu, Pfaff)
After a brief review of logic, we present two axioms for set theory, the Axiom of equality and the Axiom of set formation. Algebra of sets, relations, functions, equivalence relations, partitions, and arbitrary unions and intersections are studied with emphasis on proving the appropriate theorems about them. With set theory established on a more or less secure foundation, we proceed through a series of constructions. In succession, we build set theoretical models for the Natural Numbers, Integers, Rational Numbers (Here the students do all the work since the process is similar to what we went through for the Integers), Reals, and Complex Numbers (again, it is up to the students to prove the theorems here). The Reals are constructed using Dedekind cuts, which forces them to come to grips with reasoning with inequalities and mixed quantifiers. The Completeness Property of the Reals is proved, using sups and infs. Prerequisite: MATH 373 [DP 2001]
Math 475/675 Euclidean and non-Euclidean Geometry (Herald, Jabuka, Pfaff)
After showing dramatically how dangerous it is to reason from a picture, we introduce axioms gradually and examine their consequences. Emphasis is placed not only on the theorems of geometry, but also on metatheorems about independence. Elementary Absolute Geometry is studied rigorously, including the concepts of betweenness, coordinate systems, separation, SAS, and the Exterior Angle Theorem. After a discussion of the history of Euclid's Fifth Postulate, we introduce the Klein and Poincare models to motivate the study of Hyperbolic Geometry. After a brief digression on the completeness of the Reals and the Archimedean Property, we introduce the Hyperbolic Parallel Postulate and spend the rest of the semester coming to grips with the properties of this new world. As always, throughout, the emphasis is on proofs, how to find them and how to write them. Prerequisite: MATH 373 [DP 2001]
Math 485/685 Combinatorics and Graph Theory (Quint)
Combinatorics is the study of arrangements, patterns, designs, assignments, schedules, connections, and configurations. Graph theory is the study of networks. Together these areas constitute one of the fastest growing fields of modern mathematics. We present the basic mathematical theory of these areas, together with many applications. Topics Covered: Counting rules; generating functions; recurrence relations; inclusion-exclusion; pigeonhole principle; Ramsay theory; fundamental graph theory concepts (connectedness, coloring, planarity); Eulerian and Hamiltonian chains and circuits. Prerequisites: MATH 330 or consent of the instructor. MATH 285 is recommended. No previous background in combinatorics or graph theory assumed. [TQ 2001]
Math 486/686 Game Theory (Quint)
Game theory is the mathematical modeling and analysis of conflict situations involving more than one player. In particular, we study issues such as the existence of equilibria and the formation of coalitions in such situations. Applications will be given in economics and political science. Topics Covered: Extensive and strategic form games; Nash equilibrium; repeated games; matrix/bimatrix games; minimax theorem; TU/NTU solution; marriage, college admis-sions, and houseswapping games; core and Shapley value; power indices; NTU games.
Prerequisites: MATH 330 or consent of the instructor. Background in linear programming (MATH 487/687 or 751) would be helpful but is not required. No previous background in game theory assumed. [TQ 2001]
Math 487/687 Deterministic Operations Research (Quint)
In MATH 487/687 we cover the techniques of deterministic operations research. Topics include linear and integer programming, shortest paths in a network, project scheduling, dynamic programming, deterministic inventory theory, and nonlinear programming. Although we will study the theory of linear programming (in particular the simplex method) and also that of nonlinear programming, much of the focus of the course will be on model formulation of applied problems. Students will go "on line", using LINDO and GINO to solve business-school style "cases".
Prerequisites: MATH 330 or permission of the instructor. [TQ 2003]
Math 488/688 Differential and Difference Equations II (Formerly Math 423/623) (Pinsky) Partial differential equations are often used in various disciplines to model complicated problems. The goal of this course is two-fold: to study how to interpret a partial differential equation, and to investigate various methods one can use to solve different types of partial differential equations (analytical methods for exact solutions and approximate methods for numerical solutions).
Topics include classification of partial differential equations, interpretations of the heat equation, the wave equation and Laplace's equation, solutions by various analytical methods (separation of variables, eigenfunction expansion, the sine and cosine transforms, the Fourier transform, the Laplace transform, method of characteristics, change of coordinates) and approximate methods (explicit and implicit finite difference methods).
Prerequisite: MATH 285.
Stat 4xx/6xx
Stat 452/652 Statistics: Continuous Methods (Kozubowski, Panorska, Zaliapin)
"Statistics is the art of making numerical conjectures about puzzling questions." Is the medicine effective? What is the association between the Sierra snow pack and the clarity of Lake Tahoe? Is there a trend in inflation rate? What drives development around Reno? Why does a casino make a profit on the roulette?
In this course you will learn to choose appropriate models for real world situations, exercise these models using appropriate mathematical and computer techniques, and interpret the results in plain language. The topics covered in this course include: goodness of fit testing, methods of estimation, parametric and nonparametric approaches to correlation and multivariate regression, trend analysis, analysis of variance, analysis of categorical data. There will be a significant emphasis on hands-on statistical computations and data analysis and modeling methods using a statistical package (MINITAB). No prior programming experience is required.
Text: Devore, Jay, L. Probability and Statistics for Engineering and the Sciences, 5th Ed., Duxbury. The book will be supplemented with information available on the web.
Prerequisites: MATH/STAT 352 or STAT 467/667 or permission of instructor.
Note: STAT 452 is required for BA and BS in Mathematics, statistics option, while STAT 652 is required for MS in Mathematics, Statistics option. [TK 2005]
Stat 467/667 Statistical Theory (Kozubowski or Panorska)
Deepen your understanding of statistics. Discover the interesting mathematics behind the common statistical procedures used in practice such as estimation, testing hypotheses, and linear regression. Topics to be discussed should include multivariate probability distributions; details of point and interval estimation, including the methods of moments and maximum likelihood; derivations of common statistical tests and the corresponding power calculations; mathematical details of the method of least squares and the corresponding linear regression problems. Prerequisites: MATH 283, 330, and either MATH/STAT 352 or MATH 461/661.
Note: STAT 467 is required for BA and BS in Mathematics, Statistics option.
Math 7xx
Math 713 Abstract and Real Analysis I (Blackadar, Kumjian, Naik, Olson)
The focus of this course will be to develop the modern theory of integration of real- valued functions based on Lebesgue measure. The Riemann integral, familiar from Calculus, does not behave well with respect to limits. The Lebesgue integral does, and this makes it well-suited as a tool in Modern Analysis and Probability theory. The syllabus follows:
1. Set Theory
2. The Real Number System
3. Lebesgue Measure
4. The Lebesgue Integral
5. Differentiation
6. The Classical Banach Spaces
Text: H. L. Royden, Real Analysis, 3rd Edition. Prerequisites: Consent of the instructor. MATH 311 and 440/640 are recommended.
Math 721 Nonlinear Dynamics and Chaos I (Pinsky)
Dynamical systems theory explores modern ideas, techniques, and computer algorithms developed for modeling, analyzing and controlling the time-evolution of natural and man- made systems.
Pretend, for example, that you observe the initial state of a system modeling dynamics of connected elastic bodies or atoms in molecules, evolution of chemical reactions or competition in biology, ecology or economics, as well as problems in meteorology, hydrology or, say, laser physics. How do you predict the evolution of these kinds of
dynamical systems? Although questions of this nature have guided progress in science for hundreds of years, emergence of chaos theory was a turning point in these studies resulting in the development of new thinking across science and engineering.
In this course, we will study how to describe and analyze some of complex phenomena arising in nonlinear systems using relatively simple analytical and numerical techniques. We attempt to make a sound connection of mathematical derivations and physical intuition and comprehend the behavior of various dynamic models arising in engineering and physical sciences.
First part of this course two-semester course starts with analysis of relatively simple nonlinear systems described by second order differential equations. We show that despite their relative simplicity, these models describe complex phenomena that have no analog in linear dynamics. Next we study the synchronization and competition of nonlinear modes, nonlinear resonances, local bifurcations undergoing in continuous and discrete models of natural and engineering systems and enter the area of nonlinear wave. Prerequisite: MATH 330. MATH 285 is recommended. [MP 2003]
Math 722 Nonlinear Dynamics and Chaos II(Pinsky)
The second part of this two-semester course centered on deeper study of bifurcation phenomena leading to development of chaotic behavior. In this connection, we study bifurcation and chaos in continuous Lorenz equations and discrete dynamical systems as well as introduce fractals and Mandelbrot and Julia sets. Essential time is dedicated to analysis of bifurcation and chaotic behavior in various applied systems such as optical resonators, chemical reaction, and communication models, as well as to control of chaotic systems. Prerequisites: MATH 721. [MP 2003]
Math 731 Modern Algebra I (Blackadar, Jabuka, Naik, Kumjian)
The Essence of Pure Mathematics. Some of the deepest (as well as most useful) theorems in Analysis, Topology and Applied Mathematics rely on algebraic concepts. The fundamental concept of symmetry in Physics and the other Sciences depends in an essential way on the mathematical notion of group. We will revisit the material of abstract algebra with the hope that it will provide a higher understanding of its concepts. We hope reach the cyclic decomposition theorem by the end of the semester, which has the Jordan decomposition theorem as one of its most important consequences. Thereafter we investigate the role of groups in field extensions (i.e. Galois Theory). If there is time we will explore category theory and its foundational role in mathematics. Prerequisites: Consent of the instructor. We recommend MATH 330 and 331.
Math 751 Operations Research I - Linear Programming and Extensions (Quint)
A linear programming problem is a problem in which one is to optimize a linear function (of n variables) subject to linear constraints. We define an algorithm for such problems, called the simplex algorithm, and prove that it converges. We investigate the theory of
duality and that of sensitivity analysis. We extend the simplex algorithm so as to be able to solve integer programming problems and fractional programming problems. Finally we cover topics in nonlinear programming, such as the linear complementarity problem and Kuhn-Tucker theory. Note: The focus of this course is on theory, not applications. For the "applications" side of things, enroll in Math 487/687. Prerequisite: MATH 310, 330. [TQ 2003]
Math 752 Operations Research II – Stochastic Models (Quint)
In Math 752 we consider operations research models with a probabilistic component. In particular, we cover decision analysis, reliability theory, Markov Chains, queueing theory, and probabilistic inventory theory. We will also study some applications of these models. Prerequisites: MATH 330 and MATH 461/661, or consent of the instructor.
IMPORTANT NOTE: Math 751 is NOT a prerequisite for this course. [TQ 2001]
Math 753 Stochastic Models and Simulation (Kozubowski)
Stochastic models are used to represent random processes which evolve over time. Inventory levels are modeled by a Markov chain. Decay times for radioactive particles are modeled by a Poisson process. The number of customers waiting in line is modeled by a Markov process. The diffusion of a chemical through the water table is modeled by a Brownian motion. The flood stage of the Truckee River is modeled by a time series. In this course you will learn to choose appropriate models for real world situations, exercise these models using appropriate mathematical and computer techniques, and interpret the results in plain language. Topics to be discussed include stochastic process models with applications; analytic and computer modeling techniques for Markov chains; Poisson and Markov processes; Brownian motion and special topics. Prerequisites: MATH 330, MATH 461. [TK 2005]
Math 761 Techniques in Applied Mathematics (Olson or Telyakovskiy)
This course will serve as an introduction to mathematical techniques found in various fields of engineering and natural sciences. We will try to strike a balance between studying the mathematical aspects of the subject and dedicating an appropriate attention to empiric origins of our methods that should stimulate intuitive thinking and embrace multiple connections with important physically motivated problems.
We begin with Dimensional Analysis and Scaling and show its applications to problems from such diverse disciplines as chemical reactions, hydrodynamics, wave propagation and population dynamics, and also to mathematical modeling in certain softer fields where the explicit models are still unknown. We will follow with the perturbation techniques and consider their application to some of the problems already considered in this course. After that, we turn our attention to Calculus of Variations and subsequently to Integral Equations, Integral Transforms, and Green's function method. Prerequisites: MATH 283, 285, and 330. MATH 488/688 is desirable. [MP 2003]
Math 762 Techniques in Applied Mathematics (Olson or Telyakovskiy)
The second part of this two-semester sequence is centered on deeper study of various phenomena described by partial differential equations. In this broad area, we focus on analysis of dynamic behavior described by linear and nonlinear parabolic and hyperbolic PDEs as well as on analysis of more general wave phenomena occurred in continuous systems.
In this connection, we emphasize various perturbation and calculus of variation techniques that help reduce the complexity and furnish the mathematical analysis in a way consistent with physical intuition. We also explore connection between analytical and numerical techniques that leads to their fruitful cross-fertilizing. Prerequisites: MATH 283, 285, and 330. MATH 488/688 and 761 are desirable but not mandatory. [MP 2003]
Math 767 Advanced Mathematics for Earth Sciences
Applications of advanced mathematics for earth scientists and engineers. Includes elements of vector calculus, linear algebra, differential equations, probability, and statistics. These useful mathematical methods will be presented and applied in the context of real world problems. Co-requisite: MATH 283 or equivalent [MM 2003]
Stat 7xx
Stat 754 Mathematical Statistics (Kozubowski or Panorska)
This introduction to classical mathematical statistics is intended to cover mathematical details of the basic problems of parameter estimation and testing hypotheses. Topics to be discussed include statistical models and applications; modes of convergence used in statistics; methods of point and interval estimation, including Bayesian inference; elements of large sample theory; unbiasedness, sufficiency, and completeness; hypothesis testing, including likelihood ratio tests, Neyman-Pearson lemma, and most powerful tests; introduction to linear models and special topics. Prerequisite: MATH 311, 330, 461/661. Note: This course is required for MS in Mathematics, Statistics option. [TK 2005]
Stat 755 Multivariate Data Analysis (Kozubowski or Panorska)
In various areas of science, researchers frequently collect measurements on several variables. In this course we shall discuss basic statistical techniques for analyzing such multivariate data. Our focus will be both understanding theoretical concepts and practical implementation of the methods on real data sets. Topics to be discussed should include sample geometry and random sampling, the multivariate normal and related distributions,
estimation of the mean vector and the covariance matrix, multivariate linear regression models, principal components, factor analysis, canonical correlation analysis, discrimination and classification, and cluster analysis. Basic knowledge of multivariate calculus, linear algebra, and probability/statistics are assumed. Prerequisites: MATH 330, MATH 461/661. Co-requisite: STAT 452/652.
Note: This course is required for MS in Mathematics, Statistics option.
Stat 756 Survival Analysis
Researchers in the engineering, actuarial, and biomedical sciences are often faced with the problem of analyzing failure time data which represent times to occurrence of point events such as failure of an electronic component in an engineering study, filing of a claim of an insured unit in an actuarial setting, or recovery of a patient in a clinical trial. In most of these cases, one will not be able to observe the exact failure times of all observations due to monetary or time constraints, but will only be able to observe censored data. In this course, statistical methods that will handle these censored data will be discussed. Methods will vary from tools used to analyze data from a single population to regression tools. There will be a balance of theory and applications for a better understanding and appreciation of the concepts of survival analysis. Prerequisite: MATH 283 and MATH/STAT 352, or permission of instructor. Corequisite: STAT 452/652. [IA 2001]
Stat 757 Applied Regression Analysis
Learn the basic concepts of linear regression analysis such as least-squares estimation and statistical inferential procedures for model parameters. Methods for checking the adequacy of the model (residual analysis) as well as choosing the best model in light of the data gathered will also be discussed. [TK 2005]
Stat 758 Time Series Analysis (Zaliapin)
Time series analysis concerns random quantities that evolve over time. Practical examples include temperature, rainfall, river flows, stock market prices, interest rates, unemployment levels, electrical signals, customer demand, and population. In this course we will survey analytic and computer methods for time series analysis. We will explore both the time domain (autocorrelation) and frequency domain (spectral) approach. Prerequisite: MATH 311, MATH 330, and MATH/STAT 352, or permission of instructor. [TK 2005] | 677.169 | 1 |
Trigonometry
Retail Price:
$56.95$48.41
Product ID - 13SOSTG | Availability - Now Shipping
Want to get your teen ready for college math? Then, you need Switched-On Schoolhouse Trigonometry for grades 9-12! As a great prep course for advanced math courses, this one-semester, computer-based Alpha Omega curriculum covers topics like right angle trigonometry, trigonometric identities, graphing, the laws of sines and cosines, and polar coordinates. Pre-requisite is Algebra II. Includes quizzes and tests. Order today!Prepare your homeschool high schooler for future math courses and get him the Switched-On Schoolhouse Trigonometry elective for grades 9-12! Practical and informative, this one-semester, computer-based course covers trigonometry in clear, step-by-step lessons that will build your child's confidence in performing advanced math. Made for students who have completed Algebra II, this knowledge-building math course will show your teen how to develop trigonometric formulas and use them in "real world" applications! Plus, to make math lessons more fun, SOS has interactive, exciting multimedia tools like video clips, learning games, and animation to engage your high schooler in learning!
An innovative time-saver, Switched-On Schoolhouse offers homeschool parents a feature no textbook can—automatic grading and lesson planning! Now, you won't have to spend nights pouring over papers trying to check advanced math problems. In addition, this Christian-based SOS course has customizable curriculum, so you can always adjust math lessons to your student's learning pace. A built-in calendar and message center in this Alpha Omega curriculum also make organization a breeze. As an alternative to calculus, this trigonometry course will give your student a clear "big picture" of advanced math, as well as an understanding of how numeric, algebraic, and geometric concepts are used together to build a foundation of higher mathematical thinking. Don't wait to get your teen's mind in shape for college math! Give him a solid head start and order Switched-On Schoolhouse Trigonometry for grades 9-12 from Alpha Omega Publications today! Order | 677.169 | 1 |
The StudyMinder Homework System is an electronic student planner for Windows that helps students of all ages, from middle school and high school to college level, track their grades, prioritize assignments...
Having trouble doing your Math Homework? This program can help you master basic skills like reducing, factorising, simplifying and solving equations. Step by step explanations teach you how to solve...
WeBWorK is an web-based method for delivering individualized Homework problems. It gives students instant feedback and allows instructors to track their progress in real time. Problems are written in a...
DeadLine is a free program useful for solving equations, plotting graphs and obtaining an in-depth analysis of a function. Designed especially for students and engineers, the freeware combines graph plotting...
RekenTest is freeware educational software to practice arithmetic skills. It supports basic arithmetic operations like addition and subtraction, the muliplication tables and so on, as well as more advanced...
Do you spend hours trying to solve your math Homework step by step? Do you have difficulties with factoring, simplification or solving algebraic equations, inequalities or systems? Having trouble with...
Do you have a Homework assignment that needs the complete working out for a matrix question, and requires that it be neatly typed out? Are you having problems learning the mechanics of matrix functions in...
StudyMinder is a wonderful planner that allows students to get organized with their Homework, extra curricular classes, events, etc. The program is very intuitive and easy to use, specially designed for...
This is a Homework project for the Software Laboratory 4 course at the Budapest University of Technology and Economics. This program is developed by a team of two IT Engineering student, its goal is an...
The perfect solution for making graphing worksheets. For graphing equations in: algebra Homework, math class work, and for displaying professional graph images on a white board during classroom instruction....
This proyect will be many things. It will be not only a software program; but also your assistant and maybe your friend. You should be able to speak with her, ask her for advice, or ask her to do your...
A simple java utility to check if a number can be factored or is prime, and prints the factors if they exist. This util stemmed out of a desire to make it quicker for my wife to check our son's Homework,...
IMathAS is a web-based math assessment and Homework system. It is a course/learning management system (CMS/LMS/VLE) and testing system featuring algorithmic questions, similar to WebWork, Webassign, and...
Student Center is a web based student and teacher portal with the ability to organize Homework assignments, display news, and show other relevant information. This project includes the use of PHP5, MySQL, a...
Back in primary school, I got given alot of repetitive maths Homework that involved being a human calculator. I made this so I could scan in my maths Homework, OCR it and print out the answers. Also...
The Internet has made kids' Homework more exciting and each day one of new opportunity for small and large businesses. AllegroSurf takes your Internet connection and shares these possibilities with the...
APrintDirect lets you to print or save a customizable listing of the contents for any folder on your computer. Set numerous filters specifying which files you would like included or excluded. The processing... | 677.169 | 1 |
Data and technology have altered how we live and what students need to learn. Data analysis is now an integral component of high school Algebra courses. Fathom™ Dynamic Data software makes it easy to integrate data analysis and effective technology use into your current curriculum.
· With Fathom, teaching data analysis has never been easier. You can use this dynamic software to bridge the gap between concrete and abstract mathematical and statistical concepts by letting students see and manipulate data in a clear graphic form.
· You and your students can use real-world data to model relationships and derive algebraic functions.
· With Fathom, you can use data to deepen your students' understanding of mathematics, statistics, science, and social science.
· You and your students can quickly represent data in a variety of graphs, including bar charts, scatter plots, function plots, and histograms. This is very helpful for examining data and determining which graphs are most useful in a given situation.
· You'll find it easy to motivate your students and show them the real-world relevance of statistics and data analysis with Fathom's more than 300 included data sets, which offer real information about a variety of subjects, including education, science, the social sciences, and sports.
· Fathom provides you with a faster, more dynamic and engaging way to demonstrate statistical concepts than using transparencies or drawing on the board. Fathom works easily with your LCD projector, classroom computer, or SMART Board.
· With its friendly, drag-and-drop interface for data import, Fathom is easy to learn, so you won't have classroom downtime.
· You and your students can use the dynamic tools of Fathom to:
Plot values and functions on top of data and vary them dynamically with sliders to show your students the effects of variables :
· Build simulations that illuminate concepts from probability and statistics | 677.169 | 1 |
Introduction to specific topics in mathematics most useful for
planners.
Topics include: review of the vocabulary of mathematics; analysis
of
linear
and nonlinear functions with economic applications; the
mathematics of
finance; and descriptive statistics. The course will also include
an
introduction to the use of the computer as a tool for data
analysis and
display using spreadsheet
software.
PREREQUISITE:
A passing score on the basic mathematics entrance exam given
during
orientation week. 220A can be waived by a passing score on the
waiver
exam also given during orientation week.
A minimum passing grade on the final exam is required to pass the
course. Problem and computer assignments will count heavily in
determining the course grade. You are encouraged to work jointly
(groups of 2-4 persons) on the homework
and computer projects. Please turn in only one assignment
per
group.
Homework is due by 5:00 p.m. of the date specified. It will be
returned one week from that date. Homework which is submitted
late, but
before the week is over will be marked down a grade. Late homework
must
be submitted before the graded homework is returned. | 677.169 | 1 |
Math Spring 2007 Section 1, TR 5:45-7:10pm Science 107
Professor: PHONE: Home Page:
Andrew Diener 3213452
EMAIL ADDRESS: [email protected] OFFICE: OFFICE HOURS: 103F Science 1pm-5pm MTWR.
TEXT: Mathematics for Elementary Teachers, A Contemporary Approach, Seventh Edition, Gary L. Musser, William F. Burger and Blake E. Peterson, Wiley, 2006. COURSE CONTENT: (by catalog) This course includes concepts essential to mathematics for elementary school teaching candidates. Topics include: set theory, numbers and numeration, number theory, rational numbers and problem solving. This course does not meet the general education requirement in mathematics. Prerequisite: MATH 100 or equivalent. CALCULATOR: You must have access to the TI-83+ or TI-84+ graphing calculator on assignments and for part of each test. Tests may be in two parts, one with calculator and one without. HOMEWORK: A list of suggested homework problems will be given for each section covered. I do expect that you will attempt all homework problems. I will grade some, though not all, of these problems. We will spend some class time in groups working on these problems. Any homework must be turned in using the proper format and on time. NO late (or incorrectly formatted) homework will be accepted. Homework will account for 15% of your final grade. (100 pts) QUIZZES: There be will quizzes every week. These quizzes will be very short (approx. 10-15 min.) and will come almost directly from the homework. Each quiz will be worth 10 points and I will count 10 of them. Since I plan to give many of these quizzes (at least 12, hopefully 13) I will drop the extra ones. This does mean that quizzes cannot be made up. Quizzes will count for 15% of your final grade. TESTS: There will be three in-class exams, each exam worth 100 points, and a comprehensive final exam worth 150 points. I will not curve your test grades. If you miss a test for
any reason, you need to notify me no later than the day after the test to set up a time for a make-up. PROJECTS: I strongly encourage students to work together when studying mathematics (except on exams!) and to further this goal there will be several 3 Math 105 December 12, 2006 Name You must show all your work. Partial credit will be given. 1. The following table shows the amounts spent on reducing sizes of rst-grade through thirdgrade public school classes in a certain state. Year 1988 199
EXAM 3 Math 105 April 11, 2008 Name You must show all your work. Partial credit will be given. 1. A laptop computer currently costs $787. The price of the laptop is expected to decrease by 2.9% per year. Find a mathematical model for the price of the
EXAM 1 Math 105 November 2, 2006 Name You must show all your work. Partial credit will be given. 1. Calculate the value of H(t) = -16t 2 + 120 at t = 2, where H(t) is the height of a cliff diver above the water t seconds after he jumped from a 120 fo
QUIZ 3 Name 1. Find the slope and the x-intercept of the line 2x + 3y = 7.2. Do the row operations needed to put the following matrix into nal form and then write down the solutions (if any exist) to the system represented by the matrix. 1 0 0 2
iContents iChapter 1Keeping It In The Ballpark1.1 Studying PhysicsHow do you study for physics? Do you read your physics book the same way your read a book for literature class? Although physics and English literature are both intellectual di
Heinrich Rudolf Hertz(Redirected from Heinrich Hertz) Heinrich Rudolf Hertz (February 22, 1857 - January 1, 1894), was the German physicist for whom the hertz, the SI unit of frequency, is named. In 1888, he was the first to demonstrate the existenc
MIDTERM EXAM IISolutions Math 21D Temple-F06 Write solutions on the paper provided. Put your name on this exam sheet, and staple it to the front of your finished exam. Do Not Write On This Exam Sheet. Problem 1. (20pts) (a)Calculate the gradient f (x
MIDTERM EXAM IMath 21D Temple-F06 Write solutions on the paper provided. Put your name on this exam sheet, and staple it to the front of your finished exam. Do Not Write On This Exam Sheet.Problem 1. (15pts) Evaluate R x2y dA where R is the region | 677.169 | 1 |
Loci: Resources
Images of F
by Steve Phelps (Madeira High School)
Applet Description
This interactive Geogebra applet allows exploration of a linear transformation in terms of images of a closed figure that happens to be in the shape of the letter F initially. The Geogebra interface allows dragging of points and vectors to make for versatile explorations of basic linear algebra ideas. Suggested activities and exercises using the tool are included on page 2 of this posting and as a separate pdf file for easy printing.
Steve Phelps
Madeira High School
& GeoGebra Institute of OhioClick here or on the screen shot above to open the applet in a separate window.
Investigations
In the Images of F applet on page 1, the columns of the matrix are the elementary vectors e1 and e2. The blue figure is a pre-image initially in the shape of an F. The green figure is the image of the blue F under the transformation given by the matrix.
To answer the questions below, you can drag the tips of the elementary vector to set up the appropriate matrices. You may also need to drag the vertices of the blue F as well.
Warm Up: Set up the following matrices one at a time. Pay particular attention to the lattice points of F and to the lattice points of the image of F.
1.
\left[ \begin{array}{cc} 2 & 3 \\ 0 & 1 \end{array} \right]
2.
\left[ \begin{array}{cc} 1 & 0 \\ 3 & -1 \end{array} \right]
3.
\left[ \begin{array}{cc} 1 & 2 \\ 3 & 1 \end{array} \right]
4.
\left[ \begin{array}{cc} 2 & -1 \\ 2 & 1 \end{array} \right]
5.
\left[ \begin{array}{cc} -2 & 1 \\ 2 & -1 \end{array} \right]
6.
\left[ \begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array} \right]
Investigation 1: Drag the tips of the elementary vectors to set up the following matrices. Discuss the transformations and the resulting image of F under these matrix transformations.
transformations with matrices of the form
\left[ \begin{array}{cc} k & 0 \\ 0 & 1 \end{array} \right]
transformations with matrices of the form
\left[ \begin{array}{cc} 1 & 0 \\ 0 & k \end{array} \right]
transformations with matrices of the form
\left[ \begin{array}{cc} k & 0 \\ 0 & k \end{array} \right]
transformations with matrices of the form
\left[ \begin{array}{cc} 0 & k \\ k & 0 \end{array} \right]
transformations with matrices of the form
\left[ \begin{array}{cc} 1 & 0 \\ k & 1 \end{array} \right]
transformations with matrices of the form
\left[ \begin{array}{cc} 1 & k \\ 0 & 1 \end{array} \right]
Investigation 2: Drag the tips of the elementary vectors to set up matrices that will perform the following transformations. Pay attention to the orientation of the vectors.
Reflection over the x – axis
Reflection over the y – axis
90-degree clockwise rotation around the origin
Half-turn around the origin
90-degree counterclockwise rotation around the origin
Reflection over the line y = x
Reflection over the line y = -x
Copyright 2013. All rights reserved. The Mathematical Association of America. | 677.169 | 1 |
Acworth, GA ACTArithmetic reasoning on the ASVAB involves basic mathematical operations of real numbers and word problems related to our daily lives. Mathematics knowledge section of the ASVAB includes some more basic math, geometry, and algebraic equations. The most effective way to learn math is from studying and practicing it regularly. | 677.169 | 1 |
Descripción del producto
Descripción del producto
Packed with fully explained examples, LaTeX Beginner's Guide is a hands-on introduction quickly leading a novice user to professional-quality results. If you are about to write mathematical or scientific papers, seminar handouts, or even plan to write a thesis, then this book offers you a fast-paced and practical introduction. Particularly during studying in school and university you will benefit much, as a mathematician or physicist as well as an engineer or a humanist. Everybody with high expectations who plans to write a paper or a book will be delighted by this stable software.
Biografía del autor
Stefan Kottwitz studied mathematics in Jena and Hamburg. Afterwards, he worked as an IT Administrator and Communication Officer onboard cruise ships for AIDA Cruises and for Hapag-Lloyd Cruises. Following 10 years of sailing around the world, he is now employed as a Network and IT Security Engineer for AIDA Cruises, focusing on network infrastructure and security such as managing firewall systems for headquarters and fleet.
In between contracts, he worked as a freelance programmer and typography designer. For many years he has been providing LaTeX support in online forums. He became a moderator of the web forum latex community.org and of the site golatex.de. Recently, he began supporting the newly established Q and A site tex.stackexchange.com as a moderator.
He publishes ideas and news from the TeX world on his blog at texblog.net.
4.0 de un máximo de 5 estrellasLaTeX for beginners and beyond22 de junio de 2011
Por eloAtl - Publicado en Amazon.com
Formato:Tapa blanda
This book covers all the basics of LaTeX and then some in thirteen chapters totaling about three hundred pages. Each chapter contains a Quiz with solutions given in the Appendix. Its starts with the usual Installation chapter with the installation detailed for Windows, not other environments, maths, fonts (but not much on Unicode), long documents, hyperlinks and bookmarks to finish with a nice and useful chapter on troubleshooting, followed by online resources, answers to quizzes and an index.
The Overall, not LaTeX. It would have been nice to see LaTeX in action. Furthermore, the book has very little on Unicode. The book is about LaTeX, it does not deal at all with XeTeX. It's still useful, but you need to know it. The author uses TeXWorks on Windows. As a result, if you use a Mac and/or another editor, some pages will be useless. But then, I guess you can't really expect a book dedicated to your particular environment. On the negative side,As a disclosure, I need to say that Packt asked me if I would review the book on my site (this is an excerpt of the review). I did not get paid and Packt had no say in the review. The only perk I got was a pdf copy of the book for the review.
When I read about the publication of the book on various forums and blogs, my interest was definitely piqued: the author, Stefan Kottwitz, is a frequent and helpful contributor/moderator on TeX.SX. On the other hand I wondered if anyone would actually want to buy an introductory book to LaTeX, considering the many free tutorials and eBooks available on the Web (although there are many out-of-date ones, so beware!)
After a quick flip through the book, I felt the answer was a very firm "YES". First off, this is certainly an up-to-date book with descriptions of recent packages, and warnings about obsolete ones. While the first few chapter headings read like most other beginner's guide to LATEX, Kottwitz's approach of using complete step-by-step examples throughout the book is something seldom seen in other books or tutorials. By that I mean you don't just get the first few handful of "Hello World" examples, but for much more advanced usage scenarios as well. (BTW, The examples are based on TeXLive and TeXworks.)
Your mileage may vary, but I do feel that such a hand-holding approach (that's what my training course had been described as) -- at least in the early days of learning LaTeX -- is very reassuring. Especially so since LaTeX can be rather intimidating for people who have only used WYSIWYG word processors before. There are pop quizzes are interspersed throughout the content (answers in the appendix).
While the early chapter headings are kind of expected of any beginner's guides, they do still contain valuable nuggets. For example, the microtype package is introduced in Chapter 2, as is how to define your own macros with \newcommand. Imagine a beginner's joy at the even more beautiful typesetting afforded by microtype. And the new-found freedom of defining one's own commands for consistent typesetting of certain materials. Personally I think such tips, introduced at an early stage, would boost beginner's confidence in using LaTeX.
While some might consider the installation instructions of TeXLive and TeXworks in Chapter 1 as frivolous, I certainly welcome the instructions on how to install extra packages in Chapter 11.
Chapter 3 on designing pages is particularly useful, as this seems to be one of the most frequently asked beginner's questions these days. (At least, indicated by the fact that the post on setting page sizes and margins being the 5th all-time most favorite post on my blog.)
I also like the mention of getnonfreefonts in the chapter on fonts. Another favorite chapter of mine is that on Troubleshooting, as this is definitely one of the most important skills if one is to use (and learn!) LaTeX. And everyone who's going to write a thesis or a business report will definitely want to read Chapter 10 on large documents.
Overall, the book does cover everything a beginner should learn about LaTeX, IMHO anyway. My only nitpicks are that the LaTeX logo isn't typeset `properly' in the text; and that the LaTeXed output images seem a tad blurry in the PDF eBook version. But these are just petty nitpicks, really.
So do I recommend LaTeX Beginner's Guide for people interested in learning LaTeX? I'd say Yes. This would be a very nice addition to libraries, or as a communal copy in a research lab, so that newly registered graduate students who're not yet quite busy with their research can spend their first month learning up LaTeX with it. (You can, of course, get your very own copy; I only mention a communal copy as I know some Malaysians -- especially poor grad students -- might be reluctant to fork out about RM120 for a book. Everyone really should fork out money to buy a good book sometime, though.)
7 de 7 personas piensan que la opinión es útil
5.0 de un máximo de 5 estrellasGreat book on LaTeX - not only for beginners28 de mayo de 2011
Por Ingo Bürk - Publicado en Amazon.com
Formato:Tapa blanda
Preface: ========
The author, Stefan Kottwitz, can be found in all common (La)TeX forums as the user Stefan_K and if you're not new to LaTeX you probably already met him online. Since that was the case for me, I knew that he knows what he's talking about and so this book was a must-have for me. I read it in two days and I have to say: I am surprised and amazed. Although it's titled "Beginner's Guide", the target group definitely isn't restricted to beginners. I already wrote larger documents with LaTeX, so I wouldn't consider myself a beginner - and yet I learned a lot just by reading this book once.
Structure: ==========
The book itself is divided into 13 chapters, each being divided into smaller sections. It usually begins with explaining the topic and how to do it in general, followed by "time for action" examples, which then are explained and discussed in detail. That way, it is easy to follow his thoughts, but also to skip certain parts if you want to. I recommend reading everything though, because sometimes he gives little hints which can be really useful.
Content: ========
What can I say - amazing! From how to install a TeX distribution on your computer to how to manage even large projects this book covers everything you (or at least I) need. You really learn how to use LaTeX from scratch and, since I wasn't new to it I know that, he tells you about all the small problems you will sooner or later meet. If I would have had this book at least two years ago, I could have saved myself a lot of time using Google and forums. If you're not experienced with LaTeX, I recommend reading it carefully and really doing your own experiments rather than just copying the examples in order to really learn all the information provided and fully understand what you are doing. In any case I recommend putting little post-its on the pages which seem especially important to you - at least that's what I did and for the moment there are nine of them in my book.
I especially like that the author also talks about typography and how to write "clean" documents rather than giving instructions and commands like "here, do this and that", although this might be a little too much for a complete beginner (which just means in that case you need to read it more carefully). Also, for example, he talks about commonly used, but outdated packages or commands.
Summary: ========
I can really recommend this book, not only to beginners but also to more experienced LaTeX users. There's a lot to learn! Thanks to how it's structed you can also easily use this book to have it on your shelf and look something up in case you need to do so. | 677.169 | 1 |
Matrices are challenging, but they are really important in applied mathematics – they are a critical STEM topic. Engineers and scientists use matrices to solve challenging problems in many, many dimensions. Mathcad's matrix and graphing tools offer capabilities that can help students' explore matrices early in their school experience so that they are both prepared to use and aware of the importance of matrices. With currently available technologies matrices can be used, explored, and visualized effectively in Algebra 1 class. Systems of equations in three variables need not be avoided any longer. Matrices can be an efficient and powerful way to solve systems, with increased clarity now that we have tools to graph 3D plots.... (Show more)(Show less)
New to Mathcad Prime? This brief video illustrates how to leverage the resources on Mathcad's Getting Started tab to learn Mathcad by exploring Help and Tutorials to garner the information required for Just-in-Time learning. ... (Show more)(Show less)
Mathcad 15.0's live math capabilities provide students with timely feedback as they plot graphs, solve equations, or model data. This demonstration illustrates some useful techniques for using Mathcad to help your students be more active in directing their own learning and gain deeper understanding of mathematical concepts.... (Show more)(Show less)
Mathcad offers great features for communicating measurements, calculations, and design intent. This demonstration shows how students can use Mathcad to document and illustrate designs or solve problems in math or engineering. ... | 677.169 | 1 |
Purchasing Options
Features
Contains over 30 complete demonstrations that can be used directly in the classroom, including suggestions on how to use them
Provides the tools you need to implement your own creative ideas
Includes a CD-ROM convenient for both PC and Macintosh users that contains all of the Maple code used in the book
Presents all demonstrations in forms that will work in Maple 7, 8, or 9
Assumes no previous experience with Maple
Summary
There is nothing quite like that feeling you get when you see that look of recognition and enjoyment on your students' faces. Not just the strong ones, but everyone is nodding in agreement during your first explanation of the geometry of directional derivatives.
If you have incorporated animated demonstrations into your teaching, you know how effective they can be in eliciting this kind of response. You know the value of giving students vivid moving images to tie to concepts. But learning to make animations generally requires extensive searching through a vast computer algebra system for the pertinent functions. Maple Animation brings together virtually all of the functions and procedures useful in creating sophisticated animations using Maple 7, 8, or 9 and it presents them in a logical, accessible way. The accompanying CD-ROM provides all of the Maple code used in the book, including the code for more than 30 ready-to-use demonstrations.
From Newton's method to linear transformations, the complete animations included in this book allow you to use them straight out of the box. Careful explanations of the methods teach you how to implement your own creative ideas. Whether you are a novice or an experienced Maple user, Maple Animation provides the tools and skills to enhance your teaching and your students' enjoyment of the subject through animation.
Table of Contents
Getting Started The basic command line A few words about Maple arithmetic Comments Assigning names to results Built-in functions Defining functions Getting help and taking the tour Saving, quitting, and returning to a saved worksheet
Simple Animations Animating a function of a single variable Outline of an animation worksheet Demonstrations: Secant lines and tangent lines Using animated demonstrations in the classroom Watching a curve being drawn Demonstration: The squeeze theorem Animating a function of two variables Demonstrations: Hyperboloids Demonstrations: Paraboloids Demonstration: Level curves and contour plots
Editorial Reviews
"Putz designed this book for teachers of precalculus and first and second year calculus to provide a large number of animations to be used to illustrate various concepts of calculus. The accompanying CD-ROM contains all the described examples coded for use in class directly, along with suggestions for how to use these examples... This book will be very useful to faculty. Recommended." -CHOICE, 2004 | 677.169 | 1 |
I think you mentioned that the books used for math and science at Highlands for 7th grade and up are either out of print or hard to find, and I can't recall what you said about what you were going to put into your curriculum packages for homeschoolers. So can you tell me both:
A)What books you actually use at Highlands (or are planning to use going into the future) even if out of print
B)What you currently recommend for homeschoolers or will be putting into your curriculum packages
for both math and science, and for not only 7th and 8th grade, but high school as well.
Thank you so much for all the time you spend answering the questions on this board!
I can tell you easily what we use now for 7th/8th grade math and science, but I don't know yet what we are going to put in our packages for homeschoolers. We'll have to make that decision this summer.
Right now, at Highlands Latin, we use the College of the Redwoods pre-algebra book in 7th grade, and our teachers like it. That's an easy one for homeschoolers because you can buy it on lulu.com for $20! It's self-published by the college and very affordable. We use an out of print Lial/Miller Algebra I book (ISBN 0673188086) in 8th grade.
For science, we don't actually do science in 7th grade. We do world geography, using a course written by one of our teachers, and we add Greek, so there's no room for science. In 8th grade, we do physical science with an emphasis on pre-chemistry using another out of print book - Silver Burdett & Ginn Physical Science ISBN 0382-13472-9.
The problem with these out of print books is that there are no tests or quizzes, and solution manuals are hard to come by. I'm planning to meet with our 7th-8th grade teachers at the end of the year and see what they have to offer me, if anything!
Sorry not to have more information yet, but it's hard to nail the teachers down right now as they work to close out their year.
Tanya,
Do you think you'll have something available for 8th grade science before Fall? I'm trying to make my final choices soonish so I can work on lesson plans in June.
Right now I'm looking at either Holt Physical Science or Prentice Hall Physical Science. I'm also going to look at the book you use at Highlands- but I'm not crazy about making up my own tests. Do you think you would offer some from your school?
My goal is to have something by fall. If we go with the science book we use at school, we will definitely have the tests you need and a syllabus. I won't really know until the middle of June when the teachers are free to work with me. Check with me then, and I'll have a better idea of whether it's going to happen or not.
[QUOTE]In 8th grade, we do physical science with an emphasis on pre-chemistry using another out of print book - Silver Burdett & Ginn Physical Science ISBN 0382-13472-9.[/QUOTE]
This course is on my considering list for fall. You wouldn't happen to know how much lab equipment and any uncommon supplies are needed for this course? Even a rough estimate would be helpful. Thus far I've only used science curricula written specifically for homeschoolers, so the supplies are generally easy to find. | 677.169 | 1 |
Higher Order Derivatives
In this lesson, Professor John Zhu gives an introduction to the higher order derivatives. He explains how the 1st, 2nd, and 3rd derivative relate to one another and goes on to show you example problems.
This content requires Javascript to be available and enabled in your browser.
Higher Order Derivatives
To avoid confusion: treat each
level of derivative as brand new derivative
Higher order derivatives are
easier to solve because of eliminated terms
Higher Order Derivatives
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture. | 677.169 | 1 |
The focused ion beam (FIB) system is an important tool for understanding and manipulating the structure of materials at the nanoscale. Combining this system with an electron beam creates a DualBeam - a single system that can function as an imaging, analytical and sample modification tool. Presenting the principles, capabilities, challenges and applications of the FIB technique, this edited volume comprehensively covers the ion beam technology including the DualBeam. The basic principles of ion beam and two-beam systems, their interaction with materials, etching and deposition are all...
For many everyday transmissions, it is essential to protect digital information from noise or eavesdropping. This undergraduate introduction to error correction and cryptography is unique in devoting several chapters to quantum cryptography and quantum computing, thus providing a context in which ideas from mathematics and physics meet. By covering such topics as Shor's quantum factoring algorithm, this text informs the reader about current thinking in quantum information theory and encourages an appreciation of the connections between mathematics and science.Of particular interest are the new edition continues to serve as a comprehensive guide to modern and classical methods of statistical computing. The book is comprised of four main parts spanning the field:
Optimization
Integration and Simulation
Bootstrapping
Density Estimation and Smoothing
Within these sections, each chapter includes a comprehensive introduction and step-by-step implementation summaries to accompany the explanations of key methods. The new edition includes updated coverage and existing topics as well as new topics such as adaptive MCMC and bootstrapping for correlated data. ...
Business Mathematics focuses on transforming learning and teaching math into its simplest form by adopting learning by application approach. The book is refreshingly different in its approach, and endeavors to motivate student to learn the concept and apply them in real-life situations. It is purposely designed for the undergraduate students of management and commerce and covers wide range of syllabuses of different universities offering this course.
Complex Analysis presents a comprehensive and student-friendly introduction to the important concepts of the subject. Its clear, concise writing style and numerous applications make the basics easily accessible to students, and serves as an excellent resource for self-study. Its comprehensive coverage includes Cauchy-Goursat theorem, along with the description of connected domains and its extensions and a separate chapter on analytic functions explaining the concepts of limits, continuity and differentiability.
...
Discrete Mathematics will be of use to any undergraduate as well as post graduate courses in Computer Science and Mathematics. The syllabi of all these courses have been studied in depth and utmost care has been taken to ensure that all the essential topics in discrete structures are adequately emphasized. The book will enable the students to develop the requisite computational skills needed in software engineering.
...
The second edition of Business Statistics, continues to retain the clear, crisp pedagogy of the first edition. It now adds new features and an even stronger emphasis on practical, applied statistics that will enhance the text's ability in developing decision-making ability of the reader. In this edition, efforts have been made to assist readers in converting data into useful information that can be used by decision-makers in making more thoughtful, information-based decisionsMathematics lays the basic foundation for engineering students to pursue their core subjects. In Engineering Mathematics-III, the topics have been dealt with in a style that is lucid and easy to understand, supported by illustrations that enable the student to assimilate the concepts effortlessly. Each chapter is replete with exercises to help the student gain a deep insight into the subject. The nuances of the subject have been brought out through more than 300 well-chosen, worked-out examples interspersed across the book.
...
Linear Algebra is designed as a text for postgraduate and undergraduate students of Mathematics. This book explains the basics comprehensively and with clarity. The flowing narrative of the book provides a refreshing approach to the subject. Drawing on decades of experience from teaching and based on extensive discussions with teachers and students, the book simplifies proofs while doing away with needless burdensome textual details.
...
The new edition of this introductory graduate textbook provides a concise but accessible introduction to the Standard Model. It has been updated to account for the successes of the theory of strong interactions, and the observations on matter-antimatter asymmetry. It has become clear that neutrinos are not mass-less, and this book gives a coherent presentation of the phenomena and the theory that describes them. It includes an account of progress in the theory of strong interactions and of advances in neutrino physics. The book clearly develops the theoretical concepts from the...
Our knowledge of biological macromolecules and their interactions is based on the application of physical methods, ranging from classical thermodynamics to recently developed techniques for the detection and manipulation of single molecules. These methods, which include mass spectrometry, hydrodynamics, microscopy, diffraction and crystallography, electron microscopy, molecular dynamics simulations, and nuclear magnetic resonance, are complementary; each has its specific advantages and limitations. Organised by method, this textbook provides descriptions and examples of applications for...
This highly-regarded text provides an up-to-date and comprehensive introduction to modern particle physics. Extensively rewritten and updated, this 4th edition includes all the recent developments in elementary particle physics, as well as its connections with cosmology and astrophysics. As in previous editions, the balance between experiment and theory is continually emphasised. The stress is on the phenomenological approach and basic theoretical concepts rather than rigorous mathematical detail. Short descriptions are given of some of the key experiments in the field, and how they...
A basic problem in computer vision is to understand the structure of a real world scene given several images of it. Techniques for solving this problem are taken from projective geometry and photogrammetry. Here, the authors cover the geometric principles and their algebraic representation in terms of camera projection matrices, the fundamental matrix and the trifocal tensor. The theory and methods of computation of these entities are discussed with real examples, as is their use in the reconstruction of scenes from multiple images. The new edition features an extended introduction...
This interdisciplinary study of infinity explores the concept through the prism of mathematics and then offers more expansive investigations in areas beyond mathematical boundaries to reflect the broader, deeper implications of infinity for human intellectual thought. More than a dozen world-renowned researchers in the fields of mathematics, physics, cosmology, philosophy and theology offer a rich intellectual exchange among various current viewpoints, rather than displaying a static picture of accepted views on infinity. The book starts with a historical examination of the transformation...
The Standard Model is the most comprehensive physical theory ever developed. This textbook conveys the basic elements of the Standard Model using elementary concepts, without the theoretical rigor found in most other texts on this subject. It contains examples of basic experiments, allowing readers to see how measurements and theory interplay in the development of physics. The author examines leptons, hadrons and quarks, before presenting the dynamics and the surprising properties of the charges of the different forces. The textbook concludes with a brief discussion on the recent... | 677.169 | 1 |
Free Equation Wizard vista download
Equation Wizard 1.21You just enter an equation you see in your textbook or notebook and click one button! In an instant, you get the step-by-step solution of the equation with the found roots and the description of each step. The solution is completely automatic and does not require any math knowledge from you. Then just print the result or save it to a file.
Besides, the program allows you to simplify math expressions with one variable. Use this feature to speed up your calculations. Equation Wizard is an indispensable assistant for students at university and at school allowing them to save their time and make their learning easier.
...Learner, Veteran, Calculator, Math Pro, Math Whiz,...Math Genius, and Einstein....value and the equation to solve), choose...figure out the equation using the given...figure out the equation and solve for...
...a free Windows calculator with basic and...scientific functions. The calculator is designed with...read display. The calculator operates with a...features of this calculator include memory functions,...key (EEX). The calculator includes a link...eCalc`s online scientific calculator which supports more...
...to use scientific calculator with many advanced...complex number math, equation solving, and even...operating modes. The calculator has an attractive...disciplines. The complex math features of eCalc...Scientific Calculator allow for calculations...a constants library, equation solver (for both... | 677.169 | 1 |
Numberical Solution of Algebraic Systems Notes
Course: MATH 5485, Fall 2009 School: UCF Rating:
Word Count: 5304
Document Preview, including sound waves, water waves, elastic waves, electromagnetic waves, and so on. For simplicity, we restrict our attention to the case of waves in a one-dimensional medium, e.g., a string, bar, or column of air. We begin with a general discussion of finite difference formulae for numerically approximating derivatives of functions. The basic finite difference scheme is obtained by replacing the derivatives in the equation by the appropriate numerical differentiation formulae. However, there is no guarantee that the resulting numerical scheme will accurately approximate the true solution, and further analysis is required to elicit bona fide, convergent numerical algorithms. In dynamical problems, the finite difference schemes replace the partial differential equation by an iterative linear matrix system, and the analysis of convergence relies on the methods covered in Section 7.1. We will only introduce the most basic algorithms, leaving more sophisticated variations and extensions to a more thorough treatment, which can be found in numerical analysis texts, e.g., [5, 7, 28].
11.1. Finite Differences.
In general, to approximate the derivative of a function at a point, say f (x) or f (x), one constructs a suitable combination of sampled function values at nearby points. The underlying formalism used to construct these approximation formulae is known as the calculus of finite differences. Its development has a long and influential history, dating back to Newton. The resulting finite difference numerical methods for solving differential equations have extremely broad applicability, and can, with proper care, be adapted to most problems that arise in mathematics and its many applications. The simplest finite difference approximation is the ordinary difference quotient u(x + h) - u(x) u (x), h 4/20/07 186
c 2006
(11.1)
Peter J. Olver
One-Sided Difference Figure 11.1.
Central Difference Finite Difference Approximations.
used to approximate the first derivative of the function u(x). Indeed, if u is differentiable at x, then u (x) is, by definition, the limit, as h 0 of the finite difference quotients. Geometrically, the difference quotient equals the slope of the secant line through the two points x, u(x) and x + h, u(x + h) on the graph of the function. For small h, this should be a reasonably good approximation to the slope of the tangent line, u (x), as illustrated in the first picture in Figure 11.1. How close an approximation is the difference quotient? To answer this question, we assume that u(x) is at least twice continuously differentiable, and examine the first order Taylor expansion (11.2) u(x + h) = u(x) + u (x) h + 1 u () h2 . 2 We have used the Cauchy formula for the remainder term, in which represents some point lying between x and x + h. The error or difference between the finite difference formula and the derivative being approximated is given by u(x + h) - u(x) - u (x) = h
1 2
u () h.
(11.3)
Since the error is proportional to h, we say that the finite difference quotient (11.3) is a first order approximation. When the precise formula for the error is not so important, we will write u(x + h) - u(x) + O(h). (11.4) u (x) = h The "big Oh" notation O(h) refers to a term that is proportional to h, or, more rigorously, bounded by a constant multiple of h as h 0. Example 11.1. Let u(x) = sin x. Let us try to approximate u (1) = cos 1 = 0.5403023 . . . by computing finite difference quotients cos 1 sin(1 + h) - sin 1 . h
The result for different values of h is listed in the following table. 4/20/07 187
c 2006 Peter J. Olver
h approximation error
1 0.067826 -0.472476
.1 0.497364 -0.042939
.01 0.536086 -0.004216
.001 0.539881 -0.000421
.0001 0.540260 -0.000042
1 We observe that reducing the step size by a factor of 10 reduces the size of the error by approximately the same factor. Thus, to obtain 10 decimal digits of accuracy, we anticipate needing a step size of about h = 10-11 . The fact that the error is more of less proportional to the step size confirms that we are dealing with a first order numerical approximation.
To approximate higher order derivatives, we need to evaluate the function at more than two points. In general, an approximation to the nth order derivative u(n) (x) requires at least n+1 distinct sample points. For simplicity, we shall only use equally spaced points, leaving the general case to the exercises. For example, let us try to approximate u (x) by sampling u at the particular points x, x + h and x - h. Which combination of the function values u(x - h), u(x), u(x + h) should be used? The answer to such a question can be found by consideration of the relevant Taylor expansions u(x + h) = u(x) + u (x) h + u (x) h2 h3 + u (x) + O(h4 ), 2 6 h2 h3 u(x - h) = u(x) - u (x) h + u (x) - u (x) + O(h4 ), 2 6 u(x + h) + u(x - h) = 2 u(x) + u (x) h2 + O(h4 ). Rearranging terms, we conclude that u(x + h) - 2 u(x) + u(x - h) + O(h2 ), (11.6) h2 The result is known as the centered finite difference approximation to the second derivative of a function. Since the error is proportional to h2 , this is a second order approximation. u (x) = Example 11.2. Let u(x) = ex , with u (x) = (4 x2 + 2) ex . Let us approximate u (1) = 6 e = 16.30969097 . . . by using the finite difference quotient (11.6): e(1+h) - 2 e + e(1-h) . 6e h2 The results are listed in the following table.
h approximation error 1 50.16158638 33.85189541 .1 16.48289823 0.17320726 .01 16.31141265 0.00172168 .001 16.30970819 0.00001722 .0001 16.30969115 0.00000018
2 2 2 2
(11.5)
where the error terms are proportional to h4 . Adding the two formulae together gives
1 Each reduction in step size by a factor of 10 reduces the size of the error by a factor of 1 and results in a gain of two new decimal digits of accuracy, confirming that the finite 100 difference approximation is of second order.
4/20/07
188
c 2006
Peter J. Olver
However, this prediction is not completely borne out in practice. If we take h = .00001 then the formula produces the approximation 16.3097002570, with an error of 0.0000092863 -- which is less accurate that the approximation with h = .0001. The problem is that round-off errors have now begun to affect the computation, and underscores the difficulty with numerical differentiation. Finite difference formulae involve dividing very small quantities, which can induce large numerical errors due to round-off. As a result, while they typically produce reasonably good approximations to the derivatives for moderately small step sizes, to achieve high accuracy, one must switch to a higher precision. In fact, a similar comment applied to the previous Example 11.1, and our expectations about the error were not, in fact, fully justified as you may have discovered if you tried an extremely small step size. Another way to improve the order of accuracy of finite difference approximations is to employ more sample points. For instance, if the first order approximation (11.4) to the first derivative based on the two points x and x + h is not sufficiently accurate, one can try combining the function values at three points x, x + h and x - h. To find the appropriate combination of u(x -h), u(x), u(x +h), we return to the Taylor expansions (11.5). To solve for u (x), we subtract the two formulae, and so u(x + h) - u(x - h) = 2 u (x) h + u (x) h3 + O(h4 ). 3
Rearranging the terms, we are led to the well-known centered difference formula u (x) = u(x + h) - u(x - h) + O(h2 ), 2h (11.7)
which is a second order approximation to the first derivative. Geometrically, the centered difference quotient represents the slope of the secant line through the two points x - h, u(x - h) and x + h, u(x + h) on the graph of u centered symmetrically about the point x. Figure 11.1 illustrates the two approximations; the advantages in accuracy in the centered difference version are graphically evident. Higher order approximations can be found by evaluating the function at yet more sample points, including, say, x + 2 h, x - 2 h, etc. Example 11.3. Return to the function u(x) = sin x considered in Example 11.1. The centered difference approximation to its derivative u (1) = cos 1 = 0.5403023 . . . is cos 1 The results are tabulated as follows: sin(1 + h) - sin(1 - h) . 2h
This next computation depends upon the computer's precision; here we used single precision in Matlab.
The terms O(h4 ) do not cancel, since they represent potentially different multiples of h4 .
4/20/07
189
c 2006
Peter J. Olver
h approximation error
.1 0.53940225217 -0.00090005370
.01 0.54029330087 -0.00000900499
.001 0.54030221582 -0.00000009005
.0001 0.54030230497 -0.00000000090
As advertised, the results are much more accurate than the one-sided finite difference approximation used in Example 11.1 at the same step size. Since it is a second order 1 approximation, each reduction in the step size by a factor of 10 results in two more decimal places of accuracy. Many additional finite difference approximations can be constructed by similar manipulations of Taylor expansions, but these few very basic ones will suffice for our subsequent purposes. In the following subsection, we apply the finite difference formulae to develop numerical solution schemes for the heat and wave equations.
11.2. Numerical Algorithms for the Heat Equation.
Consider the heat equation 2u u = , t x2 0<x< , t 0, (11.8)
representing a homogeneous diffusion process of, sqy, heat in bar of length and constant thermal diffusivity > 0. The solution u(t, x) represents the temperature in the bar at time t 0 and position 0 x . To be concrete, we will impose time-dependent Dirichlet boundary conditions u(t, 0) = (t), u(t, ) = (t), t 0, (11.9)
specifying the temperature at the ends of the bar, along with the initial conditions u(0, x) = f (x), 0x , (11.10)
specifying the bar's initial temperature distribution. In order to effect a numerical approximation to the solution to this initial-boundary value problem, we begin by introducing a rectangular mesh consisting of points (ti , xj ) with 0 = x0 < x1 < < xn = and 0 = t 0 < t1 < t2 < .
For simplicity, we maintain a uniform mesh spacing in both directions, with h = xj+1 - xj = n , k = ti+1 - ti ,
representing, respectively, the spatial mesh size and the time step size. It will be essential that we do not a priori require the two to be the same. We shall use the notation ui,j u(ti , xj ) where ti = i k, xj = j h, (11.11)
to denote the numerical approximation to the solution value at the indicated mesh point. 4/20/07 190
c 2006 Peter J. Olver
As a first attempt at designing a numerical method, we shall use the simplest finite difference approximations to the derivatives. The second order space derivative is approximated by (11.6), and hence u(ti , xj+1 ) - 2 u(ti , xj ) + u(ti , xj-1 ) 2u (ti , xj ) + O(h2 ) x2 h2 ui,j+1 - 2 ui,j + ui,j-1 + O(h2 ), h2
(11.12)
where the error in the approximation is proportional to h2 . Similarly, the one-sided finite difference approximation (11.4) is used for the time derivative, and so ui+1,j - ui,j u(ti+1 , xj ) - u(ti , xj ) u (11.13) (ti , xj ) + O(k) + O(k), t k k where the error is proportion to k. In practice, one should try to ensure that the approximations have similar orders of accuracy, which leads us to choose k h2 . Assuming h < 1, this requirement has the important consequence that the time steps must be much smaller than the space mesh size. Remark : At this stage, the reader might be tempted to replace (11.13) by the second order central difference approximation (11.7). However, this produces significant complications, and the resulting numerical scheme is not practical. Replacing the derivatives in the heat equation (11.14) by their finite difference approximations (11.12), (11.13), and rearranging terms, we end up with the linear system ui+1,j = ui,j+1 + (1 - 2 )ui,j + ui,j-1 , in which = i = 0, 1, 2, . . . , j = 1, . . . , n - 1, (11.14)
k . (11.15) h2 The resulting numerical scheme takes the form of an iterative linear system for the solution values ui,j u(ti , xj ), j = 1, . . . , n - 1, at each time step ti . The initial condition (11.10) means that we should initialize our numerical data by sampling the initial temperature at the mesh points: u0,j = fj = f (xj ), ui,0 = i = (ti ), j = 1, . . . , n - 1. (11.16)
Similarly, the boundary conditions (11.9) require that ui,n = i = (ti ), i = 0, 1, 2, . . . . (11.17)
For consistency, we should assume that the initial and boundary conditions agree at the corners of the domain: f0 = f (0) = u(0, 0) = (0) = 0 , 4/20/07 191 fn = f ( ) = u(0, ) = (0) = 0 .
c 2006 Peter J. Olver2.
A Solution to the Heat Equation.
The three equations (11.1417) completely prescribe the numerical approximation algorithm for solving the initial-boundary value problem (11.810). Let us rewrite the scheme in a more transparent matrix form. First, let u(i) = ui,1 , ui,2 , . . . , ui,n-1
T
u(ti , x1 ), u(ti , x2 ), . . . , u(ti , xn-1 )
T
(11.18)
be the vector whose entries are the numerical approximations to the solution values at time ti at the interior nodes. We omit the boundary nodes x0 = 0, xn = , since those values are fixed by the boundary conditions (11.9). Then (11.14) assumes the compact vectorial form (11.19) u(i+1) = A u(i) + b(i) , where 1 - 2 A= 1 - 2 1 - 2 .. .. . . .. .. . 1 - 2 , i 0 0 . . = . . 0 i
b(i)
(11.20)
.
The coefficient matrix A is symmetric and tridiagonal. The contributions (11.17) of the boundary nodes appear in the vector b(i) . This numerical method is known as an explicit scheme since each iterate is computed directly without relying on solving an auxiliary equation -- unlike the implicit schemes to be discussed below. Example 11.4. Let us fix the diffusivity = 1 and the bar length the initial temperature profile 0 x 1, - x, 5 1 7 x - 2, x 10 , u(0, x) = f (x) = 5 5 7 1 - x, 10 x 1, 4/20/07 192
c 2006
= 1. Consider
(11.21)
Peter J. Olver
1
1
1
0.5
0.5
0.5
0.2 -0.5
0.4
0.6
0.8
1 -0.5
0.2
0.4
0.6
0.8
1 -0.5
0.2
0.4
0.6
0.8
1
-1
-1
-3.
Numerical Solutions for the Heat Equation Based on the Explicit Scheme.
on a bar of length 1, plotted in the first graph in Figure 11.2. The solution is plotted at the successive times t = ., .02, .04, . . . , .1. Observe that the corners in the initial data are immediately smoothed out. As time progresses, the solution decays, at an exponential rate of 2 9.87, to a uniform, zero temperature, which is the equilibrium temperature distribution for the homogeneous Dirichlet boundary conditions. As the solution decays to thermal equilibrium, it also assumes the progressively more symmetric shape of a single sine arc, of exponentially decreasing amplitude. In our numerical solution, we take the spatial step size h = .1. In Figure 11.3 we compare two (slightly) different time step sizes on the same initial data as used in (11.21). The first sequence uses the time step k = h2 = .01 and plots the solution at times t = 0., .02, .04. The solution is already starting to show signs of instability, and indeed soon thereafter becomes completely wild. The second sequence takes k = .005 and plots the solution at times t = 0., .025, .05. (Note that the two sequences of plots have different vertical scales.) Even though we are employing a rather coarse mesh, the numerical solution is not too far away from the true solution to the initial value problem, which can be found in Figure 11.2. In light of this calculation, we need to understand why our scheme sometimes gives reasonable answers but at other times utterly fails. To this end, let us specialize to homogeneous boundary conditions u(t, 0) = 0 = u(t, ), whereby i = i = 0 for all i = 0, 1, 2, 3, . . . , (11.22) (11.23)
and so (11.19) reduces to a homogeneous, linear iterative system u(i+1) = A u(i) .
According to Proposition 7.8, all solutions will converge to zero, u(i) 0 -- as they are supposed to (why?) -- if and only if A is a convergent matrix. But convergence depends upon the step sizes. 11.4 Example is indicating that for mesh size h = .1, the time step 4/20/07 193
c 2006 Peter J. Olver
k = .01 yields a non-convergent matrix, while k = .005 leads to a convergent matrix and a valid numerical scheme. As we learned in Theorem 7.11, the convergence property of a matrix is fixed by its spectral radius, i.e., its largest eigenvalue in magnitude. There is, in fact, an explicit formula for the eigenvalues of the particular tridiagonal matrix in our numerical scheme, which follows from the following general result. Lemma 11.5. The eigenvalues of an (n - 1) (n - 1) tridiagonal matrix all of whose diagonal entries are equal to a and all of whose sub- and super-diagonal entries are equal to b are k (11.24) , k = 1, . . . , n - 1. k = a + 2 b cos n Proof : The corresponding eigenvectors are vk = k 2k sin , sin , n n ... nk sin n
T
.
Indeed, the j th entry of the eigenvalue equation A vk = k vk reads a sin jk + b n sin (j - 1) k (j + 1) k + sin n n = a + 2 b cos k n sin jk , n
which follows from the trigonometric identity sin + sin = 2 cos + - sin . 2 2 Q.E .D.
In our particular case, a = 1 - 2 and b = , and hence the eigenvalues of the matrix A given by (11.20) are k , k = 1, . . . , n - 1. n Since the cosine term ranges between -1 and +1, the eigenvalues satisfy k = 1 - 2 + 2 cos 1 - 4 < k < 1. Thus, assuming that 0 < 1 guarantees that all | k | < 1, and hence A is a convergent 2 matrix. In this way, we have deduced the basic stability criterion = 1 k , 2 h 2 or k h2 . 2 (11.25)
With some additional analytical work, [28], it can be shown that this is sufficient to conclude that the numerical scheme (11.1417) converges to the true solution to the initialboundary value problem for the heat equation. Since not all choices of space and time steps lead to a convergent scheme, the numerical method is called conditionally stable. The convergence criterion (11.25) places a severe restriction on the time step size. For instance, if we have h = .01, and = 1, then we can only use a time step size k .00005, which is minuscule. It would take an inordinately 4/20/07 194
c 2006 Peter J. Olver
large number of time steps to compute the value of the solution at even a moderate times, e.g., t = 1. Moreover, owing to the limited accuracy of computers, the propagation of round-off errors might then cause a significant reduction in the overall accuracy of the final solution values. An unconditionally stable method -- one that does not restrict the time step -- can be constructed by using the backwards difference formula u(ti , xj ) - u(ti-1 , xj ) u + O(hk ) (ti , xj ) t k (11.26)
to approximate the temporal derivative. Substituting (11.26) and the same approximation (11.12) for uxx into the heat equation, and then replacing i by i + 1, leads to the iterative system i = 0, 1, 2, . . . , ui+1,j - ui+1,j+1 - 2 ui+1,j + ui+1,j-1 = ui,j , (11.27) j = 1, . . . , n - 1, where the parameter = k/h2 is as above. The initial and boundary conditions also have the same form (11.16), (11.17). The system can be written in the matrix form A u(i+1) = u(i) + b(i+1) , (11.28)
where A is obtained from the matrix A in (11.20) by replacing by - . This defines an implicit method since we have to solve a tridiagonal linear system at each step in order to compute the next iterate u(i+1) . However, as we learned in Section 4.5, tridiagonal systems can be solved very rapidly, and so speed does not become a significant issue in the practical implementation of this implicit scheme. Let us look at the convergence properties of the implicit scheme. For homogeneous Dirichlet boundary conditions (11.22), the system takes the form u(i+1) = A-1 u(i) , and the convergence is now governed by the eigenvalues of A-1 . Lemma 11.5 tells us that the eigenvalues of A are k = 1 + 2 - 2 cos k , n k = 1, . . . , n - 1.
As a result, its inverse A-1 has eigenvalues 1 = k 1 1 + 2 k 1 - cos n , k = 1, . . . , n - 1.
Since > 0, the latter are always less than 1 in absolute value, and so A is always a convergent matrix. The implicit scheme (11.28) is convergent for any choice of step sizes h, k, and hence unconditionally stable. 4/20/07 195
c 2006 Peter4.
Numerical Solutions for the Heat Equation Based on the Implicit Scheme.
Example 11.6. Consider the same initial-boundary value problem considered in Example 11.4. In Figure 11.4, we plot the numerical solutions obtained using the implicit scheme. The initial data is not displayed, but we graph the numerical solutions at times t = .2, .4, .6 with a mesh size of h = .1. On the top line, we use a time step of k = .01, while on the bottom k = .005. Unlike the explicit scheme, there is very little difference between the two -- both come much closer to the actual solution than the explicit scheme. Indeed, even significantly larger time steps give reasonable numerical approximations to the solution. Another popular numerical scheme is the CrankNicolson method ui+1,j - ui,j = ui+1,j+1 - 2 ui+1,j + ui+1,j-1 + ui,j+1 - 2 ui,j + ui,j-1 . 2 (11.29)
which can be obtained by averaging the explicit and implicit schemes (11.14, 27). We can write the iterative system in matrix form B u(i+1) = C u(i) + where -1 2 1+ -1 2 .. 1 . -2 .. . . . , . .. .
1 2
b(i) + b(i+1) ,
1 2 1 2
1+ -1 2 B=
1- 1 2 C=
1-
1 2
. .
.. ..
. . . . .. .
(11.30)
Convergence is governed by the generalized eigenvalues of the tridiagonal matrix pair B, C, or, equivalently, the eigenvalues of the product B -1 C, which are 1- k = 1+ 1 - cos k n k 1 - cos n 196
,
k = 1, . . . , n - 1.
(11.31)
4/20/07
c 2006
Peter5.
Numerical Solutions for the Heat Equation Based on the CrankNicolson Scheme.
Since > 0, all of the eigenvalues are strictly less than 1 in absolute value, and so the CrankNicolson scheme is also unconditionally stable. A detailed analysis will show that the errors are of the order of k 2 and h2 , and so it is reasonable to choose the time step to have the same order of magnitude as the space step, k h. This gives the CrankNicolson scheme one advantage over the previous two methods. However, applying it to the initial value problem considered earlier points out a significant weakness. Figure 11.5 shows the result of running the scheme on the initial data (11.21). The top row has space and time step sizes h = k = .1, and does a rather poor job replicating the solution. The second row uses h = k = .01, and performs better except near the corners where an annoying and incorrect local time oscillation persists as the solution decays. Indeed, since most of its eigenvalues are near -1, the CrankNicolson scheme does not do a good job of damping out the high frequency modes that arise from small scale features, including discontinuities and corners in the initial data. On the other hand, most of the eigenvalues of the fully implicit scheme are near zero, and it tends to handle the high frequency modes better, losing out to CrankNicolson when the data is smooth. Thus, a good strategy is to first evolve using the implicit scheme until the small scale noise is dissipated away, and then switch to CrankNicolson to use a much larger time step for final the large scale changes.
11.3. Numerical Solution Methods for the Wave Equation.
Let us now look at some numerical solution techniques for the wave equation. Although this is in a sense unnecessary, owing to the explicit d'Alembert solution formula, the experience we gain in designing workable schemes will serve us well in more complicated situations, including inhomogeneous media, and higher dimensional problems, when analytic solution formulas are no longer available. Consider the wave equation 2u 2u = c2 , t2 x2 0<x< , t 0, (11.32)
modeling vibrations of a homogeneous bar of length with constant wave speed c > 0. We 4/20/07 197
c 2006 Peter J. Olver
impose Dirichlet boundary conditions u(t, 0) = (t), and initial conditions u (0, x) = g(x), t We adopt the same uniformly spaced mesh u(0, x) = f (x), ti = i k, xj = j h, where 0x . (11.34) u(t, ) = (t), t 0. (11.33)
. n In order to discretize the wave equation, we replace the second order derivatives by their standard finite difference approximations (11.6), namely u(ti+1 , xj ) - 2 u(ti , xj ) + u(ti-1 , xj ) 2u (ti , xj ) + O(h2 ), 2 2 t k (11.35) u(ti , xj+1 ) - 2 u(ti , xj ) + u(ti , xj-1 ) 2u 2 (t , x ) + O(k ), x2 i j h2 Since the errors are of orders of k 2 and h2 , we anticipate to be able to choose the space and time step sizes of comparable magnitude: k h. Substituting the finite difference formulae (11.35) into the partial differential equation (11.32), and rearranging terms, we are led to the iterative system ui+1,j = 2 ui,j+1 + 2 (1 - 2 ) ui,j + 2 ui,j-1 - ui-1,j , i = 1, 2, . . . , j = 1, . . . , n - 1, (11.36)
h=
for the numerical approximations ui,j u(ti , xj ) to the solution values at the mesh points. The positive parameter ck > 0 (11.37) = h depends upon the wave speed and the ratio of space and time step sizes. The boundary conditions (11.33) require that ui,0 = i = (ti ), ui,n = i = (ti ), u(i+1) = B u(i) - u(i-1) + b(i) , where 2 2 (1 - 2 ) 2 2 (1 - 2 ) 2 .. .. B= . . 2 .. .. . .
2
i = 0, 1, 2, . . . .
(11.38)
This allows us to rewrite the system in matrix form (11.39)
, u(j)
2 2 (1 - 2 ) 198
2 u1,j j u2,j 0 . . = . , b(j) = . . . . 0 u n-2,j 2 j un-1,j (11.40)
c 2006 Peter J. Olver6.
Numerically Stable Waves.
The entries of u(i) are, as in (11.18), the numerical approximations to the solution values at the interior nodes. Note that the system (11.39) is a second order iterative scheme, since computing the next iterate u(i+1) requires the value of the preceding two, u(i) and u(i-1) . The one difficulty is getting the method started. We know u(0) since u0,j = fj = f (xj ) is determined by the initial position. However, we also need to find u(1) with entries u1,j u(k, xj ) at time t1 = k in order launch the iteration, but the initial velocity ut (0, x) = g(x) prescribes the derivatives ut (0, xj ) = gj = g(xj ) at time t0 = 0 instead. One way to resolve this difficult would be to utilize the finite difference approximation gj = u1,j - gj u(k, xj ) - u(0, xj ) u (0, xj ) t k k u1,j = fj + k gj . However, the approximation (11.41) is only accurate to order k, whereas the rest of the scheme has error proportional to k 2 . Therefore, we would introduce an unacceptably large error at the initial step. To construct an initial approximation to u(1) with error on the order of k 2 , we need to analyze the local error in more detail. Note that, by Taylor's theorem, u(k, xj ) - u(0, xj ) k 2u u c2 k 2 u u (0, xj ) = (0, xj ) , (0, xj ) + (0, xj ) + k t 2 t2 t 2 x2 where the error is now of order k 2 , and, in the final equality, we have used the fact that u is a solution to the wave equation. Therefore, we find u(k, xj ) u(0, xj ) + k u c2 k 2 2 u (0, xj ) + (0, xj ) t 2 x2 c2 k 2 c2 k 2 f (xj ) fj + k gj + = f (xj ) + k g(xj ) + (f - 2 fj + fj-1 ) , 2 2 h2 j+1 199
c 2006 Peter J. Olver
(11.41)
to compute the required values7.
Numerically Unstable Waves.
where we can use the finite difference approximation (11.6) for the second derivative of f (x) if no explicit formula is known. Therefore, when we initiate the scheme by setting u1,j = or, in matrix form, u(0) = f , u(1) =
1 2 1 2
2 fj+1 + (1 - 2 )fj +
1 2
2 fj-1 + k gj ,
(11.42)
B u(0) + k g + 1 b(0) , 2
(11.43)
we will have maintained the desired order k 2 (and h2 ) accuracy. Example 11.7. Consider the particular initial value problem utt = uxx , u(0, x) = e- 400 (x-.3) ,
2
ut (0, x) = 0,
0 x 1, t 0,
u(t, 0) = u(1, 0) = 0,
subject to homogeneous Dirichlet boundary conditions on the interval [ 0, 1 ]. The initial data is a fairly concentrated single hump centered at x = .3, and we expect it to split into two half sized humps, which then collide with the ends. Let us choose a space discretization 1 consisting of 90 equally spaced points, and so h = 90 = .0111 . . . . If we choose a time step of k = .01, whereby = .9, then we get reasonably accurate solution over a fairly long time range, as plotted in Figure 11.6 at times t = 0, .1, .2, . . . , .5. On the other hand, if we double the time step, setting k = .02, so = 1.8, then, as plotted in Figure 11.7 at times t = 0, .05, .1, .14, .16, .18, we observe an instability eventually creeping into the picture that eventually overwhelms the numerical solution. Thus, the numerical scheme appears to only be conditionally stable. The stability analysis of this numerical scheme proceeds as follows. We first need to recast the second order iterative system (11.39) into a first order system. In analogy with u(i) R 2n-2 . Example 7.4, this is accomplished by introducing the vector z(i) = u(i-1) Then B -I z(i+1) = C z(i) + c(i) , where C= . (11.44) I O 4/20/07 200
c 2006 Peter J. Olver
Therefore, the stability of the method will be determined by the eigenvalues of the coeffiu cient matrix C. The eigenvector equation C z = z, where z = , can be written out v in its individual components: B u - v = u, ( B - 2 - 1) v = 0, u = v.
Substituting the second equation into the first, we find 1 v. The latter equation implies that v is an eigenvector of B with + -1 the corresponding eigenvalue. The eigenvalues of the tridiagonal matrix B are governed by Lemma 11.5, in which a = 2(1 - 2 ) and b = 2 , and hence are or Bv = + + 1 =2 1 - 2 + 2 cos k n , k = 1, . . . , n - 1.
Multiplying both sides by leads to a quadratic equation for the eigenvalues, 2 - 2 ak + 1 = 0, where 1 - 2 2 < ak = 1 - 2 + 2 cos k < 1. n (11.45)
Each pair of solutions to these n - 1 quadratic equations, namely = ak k a2 - 1 , k (11.46)
yields two eigenvalues of the matrix C. If ak > 1, then one of the two eigenvalues will be larger than one in magnitude, which means that the linear iterative system has an exponentially growing mode, and so u(i) as i for almost all choices of initial data. This is clearly incompatible with the wave equation solution that we are trying to approximate, which is periodic and hence remains bounded. On the other hand, if | ak | < 1, then the eigenvalues (11.46) are complex numbers of modulus 1, indicated stability (but not convergence) of the matrix C. Therefore, in view of (11.45), we should require that h ck < 1, or k< , (11.47) = h c which places a restriction on the relative sizes of the time and space steps. We conclude that the numerical scheme is conditionally stable. The stability criterion (11.47) is known as the Courant condition, and can be assigned a simple geometric interpretation. Recall that the wave speed c is the slope of the characteristic lines for the wave equation. The Courant condition requires that the mesh slope, which is defined to be the ratio of the space step size to the time step size, namely h/k, must be strictly greater than the characteristic slope c. A signal starting at a mesh point (ti , xj ) will reach positions xj k/c at the next time ti+1 = ti + k, which are still between the mesh points xj-1 and xj+1 . Thus, characteristic lines that start at a mesh point are not allowed to reach beyond the neighboring mesh points at the next time step. For instance, in Figure 11.8, the wave speed is c = 1.25. The first figure has equal mesh spacing k = h, and does not satisfy the Courant condition (11.47), whereas the 4/20/07 201
c 2006 Peter J. Olver
Figure 11.8.
The Courant Condition.
second figure has k = 1 h, which does. Note how the characteristic lines starting at a 2 given mesh point have progressed beyond the neighboring mesh points after one time step in the first case, but not in the second.
4/20/07
202
c 2006 "complex analysis" integer | 677.169 | 1 |
Comprehensive Instruction
To prepare students for algebra, the mathematics curriculum must simultaneously develop conceptual understanding, computational fluency, and problem-solving skills. The development of these concepts and skills is intertwined, each supporting the other and reinforcing learning.
Teachers can help by providing students with sufficient practice distributed over time and including a conceptually rich and varied mix of problems to support their learning. In addition, teachers should encourage and support students in their efforts to master difficult mathematics content. Students who believe that effort, not just inherent talent, counts in learning mathematics can improve their performance.
Multimedia Overview
Developing Conceptual Understanding, Fluency, and Problem Solving
Use this multimedia overview to learn about the value of simultaneously teaching concepts, procedures, and problem solving; the importance of practice distributed over time in developing automaticity and improving fluency, including the use of technology-based tools; and the relationship between student beliefs about learning and mathematics performance.
(8:37 min) | 677.169 | 1 |
Buy now
Detailed description Math Essentials, Middle School Level gives middle school math teachers the tools they need to help prepare all types of students (including gifted and learning disabled) for mathematics testing and the National Council of Teachers of Mathematics (NCTM) standards. Math Essentials highlights Dr. Thompson's proven approach by incorporating manipulatives, diagrams, and independent practice. This dynamic book covers thirty key objectives arranged in four sections. Each objective includes three activities (two developmental lessons and one independent practice) and a list of commonly made errors related to the objective. The book's activities are designed to be flexible and can be used as a connected set or taught separately, depending on the learning needs of your students. Most activities and problems also include a worksheet and an answer key and each of the four sections contains a practice test with an answer key.
From the contents The Author.
Notes to the Teacher.
Section 1: Number, Operation, and Quantitative Reasoning.
Objectives.
1. Compare and order fractions, decimals (including tenths and hundredths), and percents, and find their approximate locations on a number line.
5. Generate the formulas for the circumference and the area of a circle; apply the formulas to solve word problems.
6. Generate and apply the area formula for a parallelogram (including rectangles); extend to the area of a triangle.
7. Generate and apply the area formula for a trapezoid.
8. Apply nets and concrete models to find total or partial surface areas of prisms and cylinders.
9. Find the volume of a right rectangular prism, or find a missing dimension of the prism; find the new volume when the dimensions of a prism are changed proportionally.
Practice Test.
Section 4: Graphing, Statistics, and Probability.
Objectives.
1. Locate and name points using ordered pairs of rational numbers or integers on a Cartesian coordinate plane.
2. Construct and interpret circle graphs.
3. Compare different numerical or graphical models for the same data, including histograms, circle graphs, stem-and-leaf plots, box plots, and scatter plots; compare two sets of data by comparing their graphs of similar type.
4. Find the mean of a given set of data, using different representations such as tables or bar graphs.
5. Find the probability of a simple event and its complement.
6. Find the probability of a compound event (dependent or independent). | 677.169 | 1 |
Posts Tagged 'time'
High school math is the next step in a child's life where they are taken out of the basic principles of mathematics and ushered into a world of greater depth, learning skills that they will carry with them into adulthood and hopefully exercise their critical thinking skills in a way that will help them in their career field of choice. Starting with the first year of high school, 9th grade math builds upon the basics and also introduces new components of practical math application, such as consumer math and finances.
9th Grade Math
An example of a typical 9th grade math course will include the following :
Consumer Mathematics
Applied Business Math
Pre-Algebra
Algebra 1
Geometry
Some students may excel beyond the required courses in their first year and go on to study Algebra 2, Advanced Algebra, and perhaps even venture into calculus. Every student is different and now is the time for them to truly understand and put into practice all of the equations they were taught up to transitioning from elementary education, to high school.
An important element of approaching 9th grade math, is to consider the student and what interests they have, and the goals they have for themselves in the future. If they have plans to attend a college, and what degree they wish to achieve can affect the course material they pursue while working toward their high school diploma. When looking at 9th grade math the entire four years of high school should also come into focus and contribute to what and how the student studies.
Along with learning new math expressions students will begin to learn how to use a variety of problem solving strategies. They will learn how to determine what information is needed while problem solving, and how to use multiple representation to explain situations ( numerically, algebraically, graphically, etc.) They will begin to understand how to reason, prove their reasoning mathematically, and then communicate their reasoning to others.
It is important for teachers to take the time to connect with their students on this subject. Young people who are struggling in grasping basic concepts, or did not fully learn foundational principles in elementary and middle school will have a much more difficult time transitioning into higher math. Algebra in particular can become a source of frustration without basic understanding. Parents should also be aware of where their child is at, and what they should be working on. Having teachers and parents who are willing to be involved can help a student feel more comfortable with asking questions and less inhibited in the math world.
9th grade truly is a transition from child to young adult. Diving into the world of higher math is one step closer to equipping the next generation with the tools they need to succeed. With proper support from teachers, parents, and a school that facilitates an atmosphere of learning through asking questions and hands on application, 9th grade math is one more brick in the foundation of a child's education that they can build upon for the rest of their lives. | 677.169 | 1 |
Description of Saxon Algebra 1/2: Teacher CD by Saxon
Based on Saxon's proven methods of incremental development and continual review strategies, the Algebra 1/2 program combines pre-algebra mathematics with a full pre-algebra course and an introduction to geometry and discrete mathematics.
This helped my first two children to be ready for calculus in college--they were able to test out of college algebra. My youngest child needed to go at a slower rate and had to take college Math in college. So it is great for those who have a bent for math and science. | 677.169 | 1 |
Number Theory & Abstract Algebra
Number Theory & Abstract Algebra
I'm currently taking a course, "Abstract Algebra I & Number Theory" and i'm wondering:
what is the difference between abstract algebra and number theory? the two topics seem meshed together.
i tried googling both of them and it doesn't really help. it's hard to tell the differences between the two.
can anyone give me a solid answer?
edit: i'm mostly wondering because we also have a course "Abstract Algebra II," and a course "Topics in Number Theory," both of which require "Abstract Algebra I & Number Theory" as a prerequisite. i'm only required to take one and i'd rather take the one i'm more interested in and better at.
Gosh, I have trouble seeing the similarity. Abstract algebra is the study of abstract groups, rings, fields, and such; it studies properties and how they generalize. They also classify groups and other such creatures.
Number theory is about whole numbers, divisibility, modular relations, and the like. It then uses other fields like algebra (to gain insight into integers considered as a group under addition, or a ring under addition/multiplication mod p, etc.), real analysis (generating functions, sequences, analytic approximations of the discrete), complex analysis (continuations of number-theoretical functions, special functions like zeta), combinatorics, graph theory, etc.
yes, 2 very similar topics my little half-wit friend.
I afraid I can't help you further today, I now take my grandmother to doctors.
sorry for my bad english.
love you all, love to your mothers
C to the T to the remBath
xxx
Number Theory & Abstract Algebra
Number Theory and Algebra are related.
Number Theory uses many areas of mathematics to solve problems, as CRGreathouse pointed out. Algebra seems to be one of the most popular. You'll get to learn about Fermat's Little Theorem (Euler's Theorem), which is one of the most common applications from Group Theory to Number Theory.
Jimmypoopens: I'm currently taking a course, "Abstract Algebra I & Number Theory" and i'm wondering: what is the difference between abstract algebra and number theory?
I understand his question. He has a book on Abstract Algebra that starts with the development of the integers, much of which he might also find in a Number Theory book. I suggest he look beyond the beginning chapters for his answer.
I see where he's coming from too. Remainders mod p form a field... remainders mod n in general form a group. Modular arithmetic and Abstract Algebra are essentially the same thing. How about the whole concept of the "algebraic number" too? Abstract Algebra has applications in number theory.
There's plenty of Number Theory that has nothing do do with Abstract Algebra, though. Just wiki Analytic Number Theory and you'll find plenty. | 677.169 | 1 |
Pre-Algebra Math
Pre-Algebra Math
The following options are provided for TMCC students whose math test scores have placed them below the Math 095 (beginning algebra) level. Each option covers the same pre-algebra topics preparing the student to test into and succeed in Math 095.
Currently Available Options
Option 1. Quick Review of Pre-Algebra Math
This option is for students who need only a little brushing up on their math skills. It consists of a series of self-tests with answers covering pre-algebra topics. No lessons are provided. For details, see Pre-Algebra Quick Review.
Option 2. ALEKS Online Course
This online self-paced course uses the ALEKS learning system. ALEKS provides an initial assessment of the student's math skills, an individualized curriculum based upon that initial assessment, guided practice problems, worked examples, video tutorials, and regular assessments during the course. The 18-week ALEKS access code to take this class currently costs $67.00 (if purchased online). For details, see ALEKS Pre-Algebra Course.
Option 3. Free Online Course
This course uses a free online Pre-Algebra textbook that provides lessons, worked examples, and practice problems with answers. Video lessons covering many of the course's topics and additional practice problems are also provided. For details, see Free Online Pre-Algebra Course.
Options Available Soon
Option 4. Pre-Algebra Workbook
This pencil-and-paper option is for students who do not want to work on a computer (or simply want additional practice). A workbook is provided at minimal cost to the student. The student works through assigned chapters and sections in the workbook.
Option 5. No-Credit Skills Center Class
This option serves students who want an instructor to provide guidance through the course, feedback on tests, tutoring help, etc. There are no formal lectures: students in the class select one of the four math review options above. There is a fee for the class. | 677.169 | 1 |
MATH 180 and 184 Workshops, Fall 2012
This website gives information about the weekly workshops associated
to MATH 180 and 184. These workshops are a key component of these
courses. Surveys of students done in the past indicate that the
vast majority think the workshops have helped them do better in
the course, and analysis of student performance on common final
exams supports this perception.
Different problem sets are used in the two courses, but the
structure of the workshops is identical. Problem sets and solutions
will be posted after all workshops for a given week are
complete; see the links immediately below.
Workshop Overview
The weekly workshops are 80 minutes long and start the second
week of classes. There are 12 workshops. The workshops are an important learning
element in the course. In MATH 180, workshops are worth 7.5% of the course grade; in MATH 184 they are worth 10%.
The primary activity in the workshops is working on weekly
Practice Problems, using prepared problem sets. The activity is
to be done in groups, and is to be facilitated rather than
tutored. In addition to this group work, in most workshops
students will individually work on a Problem to Hand In at the
end of the workshop.
Two TAs, one a graduate student and one an undergraduate, lead
the workshops. They do not tutor, but rather they facilitate.
Workshop Learning Goals
Actively think about and solve problems involving calculus
Interact with peers to discuss mathematics and problems involving
mathematics
Acquire and reinforce basic problem-solving skills including
reading problems carefully and writing down information contained
in the problem as a prelude to solving the problem
Workshop Details
Groups have three or four students each. They are formed at
the start of term and then generally remain intact through the
remainder of the term. Groups work on blackboards. Students in
each group take turns writing at the blackboard, and are not
sitting down and working individually.
The workshop grade component is based on group participation and
performance on the Quizzes. Effort rather than a correct answer
is the main criteria for earning thse marks. Click here
for details on how workshop grades are computed.
The 80 minutes for each workshop are usually broken down as
follows:
65 minutes for groups working on the Practice Problems
15 minutes for students working individually on the Quizzes
There will be no Quiz at the first or last workshop or at one
midterm workshop in which a survey will instead be administered.
Workshops Tips
Your instructor will cover the relevant facts in class before
you come to your weekly workshop. To take full advantage of the
workshop, review your notes from class and/or the textbook before
coming to your workshop. The workshops are your chance to practice
working on new problems, not to learn the basic material or work
on homework problems. But do try some homework problems on the
relevant material before your workshop. The better prepared you
are, the more you will get out of the workshops.
The workshops are not a replacement for homework. In addition
to attending the workshops, you must conscientiously work on
the homework your instructor assigns in order to do well in the
course. Participating fully with your group in the workshops
should make it easier for you to solve problems yourself on homework
assignments and exams.
Remember: the point of the workshops isn't getting
the final answer correct. It is the process of activelyengaging in problem solving with your peers. This process
will help you more than simply being told how to solve
problems by your instructor or TA. Mathematical problem solving
is a skill that is best learned by doing rather than watching.
Relax and make the most of this experience. It's probably
different from anything you've done in the past and may
seem daunting at first. Once you get the "hang" of
it, you'll find the workshops a rewarding, fun activity,
just like the thousands of students who've done these workshops
in past years. Who knows, maybe you'll even want to volunteer
to become a TA next year!
This website is maintained by Rajiv Gupta,
the workshop program coordinator. | 677.169 | 1 |
Spread of Numeracy in Early America
An entertaining and informative history of the birth of the American passion for numbers, tracing the history of numeracy from its origins in the Enlightenment to its flowering in mid-nineteenth century America.
A Mathematical Study of Human Thought
This monograph reports a thought experiment with a mathematical structure intended to illustrate the workings of a mind. It presents a mathematical theory of human thought based on pattern theory with a graph-based appro ...
Beyond the Numbers
A valuable guide to a successful career as a statistician A Career in Statistics: Beyond the Numbers prepares readers for careers in statistics by emphasizing essential concepts and practices beyond the technical tools p ...
Laboratories for Decision Making
Emphasizes applied Inter-disciplinary approach in business context- Provides increased motivation in classroom and linkage to other courses outside the classroom. Includes guided analysis and open-ended discussion questi ...
Enjoy a wide range of dissertations and theses published from graduate schools and universities from around the world. Covering a wide range of academic topics, we are happy to increase overall global access to these wor ...
This book provides an introduction to Galois theory and focuses on one central theme - the solvability of polynomials by radicals. Both classical and modern approaches to the subject are described in turn in order to hav ...
This well-developed, accessible text details the historical development of the subject throughout. It also provides wide-ranging coverage of significant results with comparatively elementary proofs, some of them new. Thi ...
This well-developed, accessible text details the historical development of the subject throughout. It also provides wide-ranging coverage of significant results with comparatively elementary proofs, some of them new. Thi ...
This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1820 Excerpt: ... any two ha ...
This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1820 edition. Excerpt: ...to ...
Significant research activity has occurred in the area of global optimization in recent years. Many new theoretical, algorithmic, and computational contributions have resulted. Despite the major importance of test proble ...
Sir Isaac Newton's philosophi Naturalis Principia Mathematica'(the Principia) contains a prose-style mixture of geometric and limit reasoning that has often been viewed as logically vague. In A Combination of Geometry Th ...
Unlike most elementary books on matrices, A Combinatorial Approach to Matrix Theory and Its Applications employs combinatorial and graph-theoretical tools to develop basic theorems of matrix theory, shedding new light on ...
Combinatorial chemistry represents a revolution in the way the pharmaceutical industry identifies and optimizes leads for drugs. This text attempts to provide a basic knowledge and a practical guide to perform combinator ...
The aim of this work is the definition of the polyhedral compactification of the Bruhat-Tits building of a reductive group over a local field. In addition, an explicit description of the boundary is given. In order to ma ...
Improve your algebra and problem-solving skills with A COMPANION TO CALCULUS! Every chapter in this companion provides the conceptual background and any specific algebra techniques you need to understand and solve calcul ...
Enjoy a wide range of dissertations and theses published from graduate schools and universities from around the world. Covering a wide range of academic topics, we are happy to increase overall global access to these wor ... | 677.169 | 1 |
Platonic Realms: Coping with Math Anxiety This page is a "minitext" within a larger math site authored by B. Sidney Smith, a doctoral candidate at University of Colorado, Boulder. His essay explores the social and educational roots of current attitudes toward and myths about math. He also includes some practical strategies for coping with math anxiety.
Help for Math Anxiety Authored by the mathematics staff at Middle Tennessee State University, this web page offers practical advice for tackling math anxiety. Although written for students in the traditional classroom, the six brief readings are helpful for anyone. They cover topics such as taking a math study skills inventory, how math is different from other subjects, how to study and take tests, and offers web links for additional information. | 677.169 | 1 |
MAT
106
- Math for Elementary Education I
This is the first course of a two-semester sequence which explores the mathematics content in grades K-6 from an advanced standpoint. Topics include: problem solving; functions and graphs; and numbers and operations. This course is open to elementary education and early childhood students only. | 677.169 | 1 |
The user reviews the coordinate system and basic geometry terms associated with the coordinate plane. After viewing examples of graphing objects, users can interactively test their understanding of th... More: lessons, discussions, ratings, reviews,...
The user reviews the slope and y-intercept of a line and learns how to graph a linear equation. After viewing examples, users can interactively practice determining the linear equation for each line | 677.169 | 1 |
2 years later, I understand this answer might not be helpful to you, lamdbafunctor, but for all of the other undergrads who come here and will see this, I believe Boyce and DiPrima's "Elementary Differential Equations and Boundary Value Problems" is exactly what you are looking for. The initial 4 chapter sequence this book follows (First order linear and nonlinear -> Second order linear and nonlinear -> Higher order linear and nonlinear) allows you to see the basic fundamentals being extended to more and more general cases and with very terse yet thorough and meaningful explanations through the entire way, it was a joy to read. From what I saw, it was almost like the book was written explicitly for self-study, as there is very little assumed detail. Many engineers find the downside to this book to be the almost complete lack of real-world modeling examples and such, and my response to them is that the purpose of Boyce/DiPrima is to gain a firm grounding in theory, while the purpose of other books like Edwards/Penney is to gain a firm grounding in physical/real-world applications. I am currently finishing up my first semester in Honors Diff Eq sophomore year, and I owe it almost entirely to this book. | 677.169 | 1 |
Geometry Quick Study Guide (BlackBerry) description
Boost Your grades with this illustrated quick-study guide.
Boost Your grades with this illustrated quick-study guide. You will use it from college to graduate school and beyond. Intended for everyone interested in Math and Science, particularly high school and undergraduate students.
Here are some key features of "Geometry Quick Study Guide (BlackBerry)":
· Clear and concise explanations
· Difficult concepts are explained in simple terms
· Illustrated with graphs and diagrams
· Search for the words or phrases | 677.169 | 1 |
What is WeBWorK ?
WeBWorK is an open-source web based homework system for math and sciences courses. WeBWorK is supported by the MAA (Mathematical association of America) and the NSA (National Science Foundation) and comes with a NPL (National Problem Library) of over 20,000 homework problems. Webwork can be used for college algebra, discrete mathematics, probability and statistics, single and multivariable calculus, differential equations, linear algebra and complex analysis. | 677.169 | 1 |
News
Key Stage 5 Mathematics
Why choose Mathematics A/AS Level?
'The highest form of pure thought is in mathematics.' (Plato)
Mathematics A Level will help your understanding of mathematics and mathematical processes. It can make sense of 'real world' problems and help develop your ability to analyse and refine a model that describes a real life situation. It can boost your confidence and self-esteem and give you great satisfaction when you crack a problem. Mathematics can help you communicate effectively, both with written work and through discussing concepts with others. You can acquire new IT skills through the use of graphical calculators and graphing computer packages. But perhaps most importantly, you will study an enjoyable and rewarding subject that is both relevant and useful to your life and your future career.
Your modules
Post 16 Mathematics is divided into two parts: Core and Applied.
Core Maths, to some extent, builds on topics covered in GCSE Maths including geometry with co-ordinates, sequences, trigonometry and vectors. It also introduces the new topic of calculus, which involves gradients of curves and areas under curves.
Applied Maths is divided into two areas: Statistics and Mechanics. Statistics is the study of the use of data, how to set up appropriate models for sets of data, estimating values in a population by using a sample and probability. Mechanics is the study of forces and of movement.
On your marks...
Year 12
Raw Score Max Mark
UMS
Examination
Core 1
75
100
1.5hr Non-Calc
Core 2
75
100
1.5hr Calc
Statistics 1
75
100
1.5hr Calc
Year 13
Core 3
75
100
1.5hr Calc
Core 4
75
100
1.5hr Calc
Mechanics 1
75
100
1.5hr Calc
To Put this in Perspective... overall in your A Level if you achieve an A overall and you average 90% in your A2 modules.
We follow the Edexcel ( scheme of learning. Click on the following link to access more information on the scheme of learning, formula booklet and support materials:
Who takes this course?
What skills will I learn?
All sorts of skills, relevant to your life and the other subjects that you study:
Logical reasoning
You will be able to tackle problems mathematically and analyse and refine models that you produce
Communication skills, both through written and oral explanations
IT skills will improve as you use computer software and graphical calculators
Increased responsibility for your own learning and gain a deeper understanding of mathematical problems.
What could this lead to in the future?
Mathematics is one of those subjects that can fit in with many things you may want to do in the future. It is especially vital if you want to study a Mathematics, Physics, Chemistry or Engineering based course at Higher Education.
How will this fit into my life?
Students who take Mathematics often also study from a wide range of subjects such as Geography, Biology and Business Studies and allows you to gain a non-arts/humanities qualification.
What do I do now?
Talk to your Mathematics teacher and get some advice as to whether the course could be right for you. Making an appointment to see your school careers advisor is also a good idea. | 677.169 | 1 |
FUGP - Fungraph - Graphs of mathematical functions - 5 types of graphs:- Single - Piecewise - Parametric - Pola and Multiple - Print and copy graph into clipboard - 15 preset examples - Easy to use - User's manual in PDF format - At home or in the classroom, FunGraph is suitable both for learning and for teaching. | 677.169 | 1 |
What can you do to help students learn the advanced math that is required in so much of today's industries and technologies? What helpful insights come from cognitive science, comparative anthropology, and educational psychology? | 677.169 | 1 |
Matt's Tutoring Blog very beginning -- the diagnostic test, of all things! -- of... Introduction (page 23) to Manhattan's Strategy Guides for the...
The most lasting way to improve your vocabulary is to learn new words (1) in context (by looking up unknown words when you read and keeping a journal of their definitions) and (2) in thematic groups -- NOT by memorizing huge lists of unrelated words. These are some of the resources I use with my students; feel free to comment to add your own favorite vocabulary book!
Most students taking the SAT, GRE, or GMAT know their algebra fairly well, but many find they can't complete all the problems in the allowed time. Why? It's NOT because those students are just naturally slow: it's because they're doing more work than they need to! It's not their speed but their very approach --- the very way they conceive of the process of problem-solving --- that's flawed. To ace the math sections of standardized tests, you have to learn how to attack problems in new ways so that you get the right answers by doing as little work as possible! (Part of the reason so many students...
Many first-year calculus students fall into a common trap: they tend to make bad assumptions about how functions behave. In particular, they tend to think all functions are "nice," in the sense of easy to draw and understand -- because most of the pictures their teachers draw in school to illustrate examples tend to be of nice, familiar functions they are comfortable working with, like polynomials. But functions, in general, are extremely unwieldy, and to truly master differential calculus, you have to learn to be on guard against making simplifying assumptions: what we often imagine to be the...
Of the vast amount of math taught in high school, combinatorics is usually the most baffling for students. In my ten years of teaching, I've never had a student who felt totally confident about counting problems -- I myself didn't feel I really understood them until I went to college! -- and the most typical reaction to them is immediate fear or frustration: students often give up as soon as they see one, before they even attempt a solution. Why? Probably because many high school math teachers don't do a good job of explaining the basic concepts with concrete examples; instead, they often present...
Many of my students preparing for the SAT, GRE, and GMAT have decent algebraic intuition when it comes to EQUATIONS, but most are much weaker when it comes to INEQUALITIES.
On the one hand, this is entirely natural: inequalities capture less information than equations -- they establish merely a relation between two quantities, rather than their equivalence -- so they are inherently trickier to think about. But on the other hand, it's crucial to have a very solid grasp of how inequalities work to do well on the SAT, GRE, and especially the GMAT (which tends to love data sufficiency questions...
Many of my students preparing for the GRE or GMAT have decent algebraic skills, but most have trouble with statistical reasoning --- for a variety of reasons. Some have never had statistics; others have been away from it for years. In either case, it's crucial to get up to speed on the basics!
To get a sense of how prepared you are for some of the more challenging statistics questions on the GRE and GMAT, check out the following worksheet I've developed. When you work with me, you'll gain exactly the skills you need to ace these and similar problems --- you'll learn to complete this entire...
Matt L.
Matt L.
passed a background check on
2/20/2013. You may run an updated background check on
Matt
once you send an email. | 677.169 | 1 |
Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more.
Course Hero has millions of course specific materials providing students with the best way to expand
their education.
CS034Intro to Systems ProgrammingDoeppner & Van HentenryckLab 6Out: Wednesday 9 March 2005 What youll learn.Modern C+ comes with a powerful template library, the Standard Template Library, or STL. The STL is based on the independent concepts
MAT125.R92: QUIZ 0SOLUTIONSNo score will be assigned for this quiz. The graph of the function f (x) is given below:11(a) Determine the domain and range of f (x). Domain: 4 < x < 1 and 1 < x < 4 (or, in other notations, (4, 1) and (1, 4). We
Chemistry 112B: Organic Chemistry Winter 2008 Professor Rebecca Braslau Assigned Homework Problems The following problems are required, and must be turned in. Problems are to b e done without looking at the answers as much as possible, then corrected
1 Problem 37 section 4.1. We have the situation shown in the gure, where v is the velocity, (xb , yb ) are the coordinates of the runner, xa is the x-coordinate of the runners friend (we do not show the y -coordinate of the runners friend since it is
MAT123 - Introduction to calculus Second Practice Midterm The Second Midterm will be on Tuesday 11/11 at 8:30pm at Harriman 137. Important: check the webpage to get a copy of Second Midterm of Fall 2007! Question 1. Compute: (a) log2 (16) (b) ln e3 +
Math 127 - S2008 Practice Test for the Final Examination1. Show that the function y =Ccos(x) x2is a solution of the dierential equationx2 y + 2xy = sin(x). For what value of C does the solution satisfy the initial condition y(2) = 0? 2. Find th
Answers for HW 9 3. Usually, one may compute the 1st, 2nd and 3rd derivatives of f at to get the 6 T3 . For this problem, one may also use sin(y + z ) = sin y cos z + cos y sin z to get the whole Taylor series of f. Just take y = x and z = .(Sinc
Math 127 - Spring 2008 Practice for Second Examination1. Find the interval of convergence for the power series(1)n+1n=2(x 6)2n . n12 3n2. Find the MacLaurin series for the function ex 1 f (x) = . x3 3. Find the sum of the innite series 2 3
Calculus Early Exam February 5, 2003Instructions: The exam consists of 15 multiple choice questions. You have 90 minutes to answer all fteen questions. Be sure to record your answers on the opscan form. You are not allowed to use any books, notes, o
MAT131 Spring 2003 Midterm II SolutionProblem Score Max 21 12 8 1 2 3 4 5 6 TotalUse of calculators, books or notes is not allowed. Show the all steps you made to find the answers. Write carefully, points may be taken off for meaningless stateme
MAT 331: Mathematical Problem Solving with ComputersStony Brook, Fall 2008General Information: This course serves as an introduction to computing for the math student. After a general introduction to the use of the computers, we will turn to more
Natasha Tuskovich Friday, June 12, 2009E. coli Bio-ThermometerIntroduction A biological thermometer would ideally be a simple, accurate and easily observable register of the organisms environment. A basic version is comprised of E. coli cells that
Math 331, Fall 2008, Problems1. Compute IFS parameters and the similarity dimension of the following fractal.1.00.750.50.250.0 0.0 0.25 0.5 0.75 1.02. (a) Find the IFS parameters to generate atractor of the Picture: a right gasket of side
Chapter 5 A turtle in a fractal garden1 Turtle GraphicsImagine you have a small turtle who responds to certain commands like move forward a step, move back a step, turn right, and turn left. Imagine also that this turtle carries a pen (or just lea
MAT331 Exercises, Fall 0812.4Write a procedure in Maple that counts the frequency of letters in a string of text. For example, here is what it looks like when I use mine: freqs("time flies like an arrow, fruit flies like a bananna."); [" ",9], [ | 677.169 | 1 |
Each of our math seminars is designed as an intensive six day course with 39 hours of instruction time and is offered either in the summer or winter break. As of today we have offered the following mathematics seminars: | 677.169 | 1 |
Calculus AB: First-Time (math 1247)
AB Calculus first timers is intended for those individuals who have not taught AP* Calculus or have been teaching AP* Calculus for less than three years. The course will review concepts of Calculus while focusing on the requirements for students to be successful on the Calculus AB exam. Much of the week will be spent looking at previous AP* exams and working the problems so that participants will get a sense of how the scoring works and how to help students earn the maximum number of points. The course will focus on the rule of four, looking at Calculus from numerical, graphical, algebraic, and verbal perspectives.
Rose Gundacker taught AP* Calculus, both AB and BC, at Rosemount High School for 20 years before retiring in 2008. Since then she has continued teaching part-time either at St. Olaf College or the University of St. Thomas. Her experience with the AP* program goes back to 1998 when she started as a reader. After 6 years as a reader, Rose spent 6 years as a table leader and then 2 years as both a table leader and a member of a question team. Last year was her first year as a question leader and she will continue in this role for the 2013 Reading. Rose Gundacker has been an AP* consultant since 2000 and has presented summer workshops and one-day workshops at various sites throughout the Midwest | 677.169 | 1 |
Peer Review
Ratings
Overall Rating:
Seeing Math™ has developed interactive software tools to clarify key mathematical ideas in middle and high school mathematics. Each interactive provides a real-time connection between representations of the mathematics (symbolic, graphical, etc.), so that changes in one representation instantly cause changes in the other. The eight applets include: Qualitative Grapher, Piecewise Linear Grapher, Linear Transformer, Function Analyzer, Quadratic Transformer, System Solver, Plop It!, and Proportioner.
Learning Goals:
To illustrate and reinforce key mathematical ideas for teachers and students of algebra. To accelerate learning and enhance comprehension of difficult concepts.
Target Student Population:
Middle school and high school students; beginning college students can also benefit
Prerequisite Knowledge or Skills:
General knowledge of functions, equations, and graphs; mean, median and mode
Type of Material:
Interactive Java applet
Recommended Uses:
Classroom demo; student exploration or enrichment
Technical Requirements:
Java-enabled Web browser
Evaluation and Observation
Content Quality
Rating:
Strengths:
The applets are well-designed and implemented. Each includes a detailed Sample Activity, and all but the Quantitative Grapher include a User Guide and a sample Warm Up activity. Some also include a section on Frequently Asked Questions.
Concerns:
None
Potential Effectiveness as a Teaching Tool
Rating:
Strengths:
The materials are well-written and the applets are effective. As noted by the authors, the applets are designed to illustrate and reinforce key mathematical ideas for teachers and students of algebra. Seeing immediate feedback to changes in parameters is most informative and can easily be related to textbook and/or classroom presentations. The Linear and Quadratic Transformers, in particular, allows students to analyze the nature of their respective functions. A nice feature is that they allow linear and quadratic functions to be entered and manipulated in multiple forms (e.g., polynomial, vertex, and root forms for a quadratic functions). Plop It! provides a nice analysis of mean, median and mode. The Warm Ups and Sample Activities are quite extensive and provide ready-to-use examples for both teachers and students.
Concerns:
The System Solver and Proportioner applets are somewhat complicated and their goals and effectiveness are not immediately obvious from the applet interface; their user guides are necessary in order to make effective use of the applets.
Ease of Use for Both Students and Faculty
Rating:
Strengths:
The applets may be downloaded for use offline.
The User Guides open in new windows and are thus easy to use while running the interactive applets.
The FAQ sections are designed to address known issues in usage.
No technical difficulties were encountered in the use of this site
Concerns:
Some of the applets are a bit complicated and their interfaces are not immediately obvious (viz., System Solver and Proportioner); however, the user guides are thorough and complete. | 677.169 | 1 |
CAROL SCHWAB @ webster university - math&computer science
Introduction to DPGraph software
What is DPGraph?
Over one million mathematicians, physicists, teachers, and students use DPGraph.
The world's most powerful software for math and physics visualization. Create beautiful, interactive, dynamic, photorealistic 2D, 3D, 4D, 5D, 6D, 7D and 8D graphs. Includes hundreds of examples contributed by users from around the world. Used for algebra, geometry, trigonometry and general physics, through multivariable calculus, field theory, quantum mechanics and gravitation. Use time and color as extra dimensions (to create motion or encode momentum, for example). Use the scrollbar to vary parameters in realtime, to slice through graphs, or to vary transparency. Programmed entirely in assembly language for maximum speed. Graph functions, equations, conic sections, planes, spheres, toruses, parametric curves and surfaces, implicit equalities and inequalities, volume intersections, volumes of integration, vector fields, surfaces of revolution, equipotential surfaces, and much more, in rectangular, polar, cylindrical, or spherical coordinates.
To download DPGraph, click on the Access DPGraph link. Follow the instructions.
*For Windows XP, 2000, NT, and 98 Operating Systems only!
Sample DPGraphs to download
Click on the name of graph icon below to download/open the .DPG file. Once downloaded, start DPGraph and OPEN the file by navigating to its location.
You can set DPGraph's parameters to do 2D graphs, too. This graph lets you use DPGraph's scrollbar to change the variables A, B, and C to explore the classic sine curve y=a sin(bx+c) using GRAPH3D((Y = A*SIN(B*X+C), X=0, Y=0)).
2D Level Curves
As with 3D, DPGraph can display 2D level curves. This is handy for viewing everything from complex variable conformal maps to the characteristic curves of partial differential equations. Here's a snapshot from a movie of equipotential lines "flowing" along streamlines from the source on the right to the sink on the left. You can use DPGraph's scrollbar to adjust A (the distance of the source or the sink from the origin) and C (the strength of the source or the sink from the origin).
Slice
One way to make a graph move is to use DPGraph's scrollbar to slice through it in the x, y, or z directions in real time. Here's a movie of the slicer sweeping through the previous level surfaces, showing what the inside of the graph looks like. | 677.169 | 1 |
0618372210
9780618372218 to mathematics, along with the integration of graphing calculators and Excel spreadsheet explorations, exposes students to the tools they will encounter in future careers.A wealth of pedagogy includes the following distinctive features: detailed Worked-out Examples with Annotations help students through more challenging concepts; Practice Problems are offered to help students check their understanding of concepts presented in the examples; Section Summaries briefly restate essential formulas and key concepts; Chapter Summary with Hints and Suggestions unify chapter themes, give specific reminders, and reference problems in the review exercises suitable for a practice test; and Cumulative Review Exercises appear at the end of groups of chapters to reinforce previously learned concepts and skills. «Show less... Show more»
Rent Finite Mathematics 2nd Edition today, or search our site for other Berresford | 677.169 | 1 |
Normal 0 false false false MicrosoftInternetExplorer4 The goal ofElementary Algebra: Concepts and Applications,7e is to help todayrs"s students learn and retain mathematical concepts by preparing them for the transition from "skills-oriented" elementaryConcept Reinforcementexercises, Student Notes that help students avoid common mistakes, and Study Summaries that highlight the most important concepts and terminology from each chapter. For all readers interested in elementary algebra.
Table of Contents
Introduction to Algebraic Expressions
Introduction to Algebra
2
(11)
The Commutative, Associative, and Distributive Laws
13
(7)
Fraction Notation
20
(10)
Positive and Negative Real Numbers
30
(9)
Addition of Real Numbers
39
(7)
Subtraction of Real Numbers
46
(8)
Multiplication and Division of Real Numbers
54
(9)
Exponential Notation and Order of Operations
63
(17)
Study Summary
74
(1)
Review Exercises
75
(3)
Test
78
(2)
Equations, Inequalities, and Problem Solving
Solving Equations
80
(9)
Using the Principles Together
89
(8)
Formulas
97
(7)
Applications with Percent
104
(10)
Problem Solving
114
(13)
Solving Inequalities
127
(8)
Solving Applications with Inequalities
135
(13)
Study Summary
143
(1)
Review Exercises
144
(2)
Test
146
(2)
Introduction to Graphing
Reading Graphs, Plotting Points, and Scaling Graphs
148
(11)
Graphing Linear Equations
159
(10)
Graphing and Intercepts
169
(8)
Rates
177
(10)
Slope
187
(14)
Slope--Intercept Form
201
(8)
Point--Slope Form
209
(17)
Study Summary
219
(1)
Review Exercises
219
(3)
Test
222
(1)
Cumulative Review: Chapters 1--3
223
(3)
Polynomials
Exponents and Their Properties
226
(9)
Polynomials
235
(9)
Addition and Subtraction of Polynomials
244
(9)
Multiplication of Polynomials
253
(8)
Special Products
261
(9)
Polynomials in Several Variables
270
(9)
Division of Polynomials
279
(5)
Negative Exponents and Scientific Notation
284
(16)
Study Summary
294
(1)
Review Exercises
295
(3)
Test
298
(2)
Polynomials and Factoring
Introduction to Factoring
300
(8)
Factoring Trinomials of the Type x2 + bx + c
308
(9)
Factoring Trinomials of the Type ax2 + bx + c
317
(9)
Factoring Perfect-Square Trinomials and Differences of Squares
326
(8)
Factoring: A General Strategy
334
(6)
Solving Quadratic Equations by Factoring
340
(8)
Solving Applications
348
(16)
Study Summary
359
(1)
Review Exercises
360
(1)
Test
361
(3)
Rational Expressions and Equations
Rational Expressions
364
(7)
Multiplication and Division
371
(6)
Addition, Subtraction, and Least Common Denominators
377
(10)
Addition and Subtraction with Unlike Denominators
387
(9)
Complex Rational Expressions
396
(6)
Solving Rational Equations
402
(8)
Applications Using Rational Equations and Proportions
410
(20)
Study Summary
423
(1)
Review Exercises
424
(2)
Test
426
(1)
Cumulative Review: Chapters 1--6
427
(3)
Systems and More Graphing
Systems of Equations and Graphing
430
(7)
Systems of Equations and Substitution
437
(7)
Systems of Equations and Elimination
444
(9)
More Applications Using Systems
453
(11)
Linear Inequalities in Two Variables
464
(5)
Systems of Linear Inequalities
469
(3)
Direct and Inverse Variation
472
(12)
Study Summary
479
(1)
Review Exercises
480
(2)
Test
482
(2)
Radical Expressions and Equations
Introduction to Square Roots and Radical Expressions
484
(8)
Multiplying and Simplifying Radical Expressions
492
(6)
Quotients Involving Square Roots
498
(5)
Radical Expressions with Several Terms
503
(5)
Radical Equations
508
(7)
Applications Using Right Triangles
515
(8)
Higher Roots and Rational Exponents
523
(11)
Study Summary
529
(1)
Review Exercises
530
(2)
Test
532
(2)
Quadratic Equations
Solving Quadratic Equations: The Principle of Square Roots
534
(6)
Solving Quadratic Equations: Completing the Square
540
(6)
The Quadratic Formula and Applications
546
(10)
Formulas and Equations
556
(6)
Complex Numbers as Solutions of Quadratic Equations
562
(4)
Graphs of Quadratic Equations
566
(6)
Functions
572
(19)
Study Summary
583
(1)
Review Exercises
584
(2)
Test
586
(1)
Cumulative Review: Chapters 1--9
587
(4)
Appendix A: Factoring Sums or Differences of Cubes
591
(3)
Appendix B: Mean, Median, and Mode
594
(4)
Appendix C: Sets
598
(5)
Table 1: Fraction and Decimal Equivalents
603
(1)
Table 2: Squares and Square Roots with Approximations to Three Decimal Places | 677.169 | 1 |
Book Description: Your guide to a higher score on the Praxis II?: Mathematics Content Knowledge Test (0061)Why CliffsTestPrep Guides?Go with the name you know and trustGet the information you need--fast!Written by test-prep specialistsAbout the contents:Introduction* Overview of the exam* How to use this book* Proven study strategies and test-taking tipsPart I: Subject Review* Focused review of all exam topics: arithmetic and basic algebra, geometry, trigonometry, analytic geometry, functions and their graphs, calculus, probability and statistics, discrete mathematics, linear algebra, computer science, and mathematical reasoning and modeling* Reviews cover basic terminology and principles, relevant laws, formulas, theorems, algorithms, and morePart II: 3 Full-Length Practice Examinations* Like the actual exam, each practice exam includes 50 multiple-choice questions* Complete with answers and explanations for all questionsTest Prep-Essentials from the Experts at CliffsNotes? | 677.169 | 1 |
1. Discuss and apply combinatorial properties of sets as well as objects constructed from them (e.g., pigeonhole principle, number of functions of a certain type between two finite sets).
2. Relate the study and properties of graphs to computational applications.
3. Discuss, apply and prove the correctness of various algorithms and results on graphs.
4. Discuss the application of appropriate algebraic operations to properties of graphs as well as the extension of applications by suitable interpretation of algebraic operations (various interpretations of matrix multiplication). | 677.169 | 1 |
Matrices are challenging, but they are really important in applied mathematics – they are a critical STEM topic. Engineers and scientists use matrices to solve challenging problems in many, many dimensions. Mathcad's matrix and graphing tools offer capabilities that can help students' explore matrices early in their school experience so that they are both prepared to use and aware of the importance of matrices. With currently available technologies matrices can be used, explored, and visualized effectively in Algebra 1 class. Systems of equations in three variables need not be avoided any longer. Matrices can be an efficient and powerful way to solve systems, with increased clarity now that we have tools to graph 3D plots.... (Show more)(Show less)
New to Mathcad Prime? This brief video illustrates how to leverage the resources on Mathcad's Getting Started tab to learn Mathcad by exploring Help and Tutorials to garner the information required for Just-in-Time learning. ... (Show more)(Show less)
Mathcad 15.0's live math capabilities provide students with timely feedback as they plot graphs, solve equations, or model data. This demonstration illustrates some useful techniques for using Mathcad to help your students be more active in directing their own learning and gain deeper understanding of mathematical concepts.... (Show more)(Show less)
Mathcad offers great features for communicating measurements, calculations, and design intent. This demonstration shows how students can use Mathcad to document and illustrate designs or solve problems in math or engineering. ... | 677.169 | 1 |
Colmar Excel Algebra II:
Algebra 2 is a course designed to build on algebraic and geometric concepts. It develops advanced algebra skills such as systems of equations, advanced polynomials,......The students needed to share pencils and paper in order to frugally use the school?s scant supplies. However, most amazingly, the school had no textbooks (a fact that most teachers complained about), but she had each student make his or her own textbook; therefore, the class had forty-some textb... | 677.169 | 1 |
College Algebra covers all of the topics generally taught in a one-term college algebra course. It is completely Web-based, and is presented in a format that is optimal for online learning, whether you teach a fully online, hybrid, or traditional course.
Student Guide and Syllabus
The Student Guide clearly explains how the course content is organized so that students can focus on learning, not on trying to figure out where to find information.
Instructors customize the syllabus to fit their course, adding information such as a class schedule with important dates, any additional requirements or resources, and grading policies.
Chapter Pretests
Students take a pretest prior to beginning each chapter in order to self-assess their current knowledge of the chapter topics and help determine where to focus their efforts as they go through the chapter. Feedback pointing students to the relevant section for each question is provided. The pretest is delivered through and graded by the course management system.
Lessons, Animations with Audio, Interactive Exercises, Practice Problems, Videos and Section Quizzes
Each chapter is broken into sections. Within each section are lessons that contain animations with audio, interactive exercises, practice problem and answer sets, videos, and quizzes.
Each section begins with a list of specific learning objectives. Lessons give students a thorough, straightforward presentation of the material, broken into small "chunks" of content. There is no extraneous information, allowing students to focus on the key elements of the specific concepts being conveyed. Equivalent to a traditional textbook explanation and lecture presentation, lessons allow students to set their own pace as they move through the material.
Within lessons, extensive use of animation helps students visualize difficult concepts and problem-solving steps. Some lessons also contain interactive examples that guide students step-by-step through problem solving, with hints for completing each step. Students are able to consider each step before requesting a hint.
Many sections contain interactive exercises (Java applets) that help students take their learning one step further.
Problem solving tutorial videos (with audio) for every section allow students to watch an instructor hand-write and explain the solution to selected practice problems.
Practice problems and a complete answer set are provided for each section. Students work through problems related to the lesson. Answers are available, most with step-by-step solutions, for students to self-assess their progress and identify areas of difficulty.
NEW! Algorithmically generated problems with hints and full solutions are provided for each section.
Automatically-graded homework problems are assigned for each section. This allows the instructor to monitor individual student progress on a regular basis. It is delivered through and graded by the course management system.
Posttests
A cumulative posttest is provided for students' self-assessment, allowing them to seek help where needed before the chapter exam. Feedback pointing students to the relevant section for review is provided for each question. It is delivered through and graded by the course management system.
Chapter Exams
Two versions of a comprehensive end-of-chapter exam are available and delivered through and graded by the course management system. Also, a NEW algorithmic testing option is available for chapter, midterm, and final exams.
Test Banks
In addition to two versions of a Chapter Exam for each module, test banks are provided for each chapter. You can use these test banks to modify existing assessments, and/or to create midterm and final exams.
Course Management
We have partnered with BlackboardTM, eCollegeTM, and AngelTM, the leading content management systems in higher education, to provide you with the best assessment, communication, and classroom management tools widely available. | 677.169 | 1 |
MATHEMATICS LEARNING OVERVIEW 2010-2011
AUTUMN TERM 1
6th September-22nd October
AUTUMN TERM 2
1st November-17th December
During the 1st term Foundation students will study
Area and perimeter of 2 D shapes (e.g. trapezium, parallelogram and triangles) and compound shapes and transformations. They will revisit areas such as simplifying, factorising and linear graphs in algebra.
Higher students will study Graphs of different functions; solve equations using trial and error method.
At end of each topic students are assessed on a 60 minute end of chapter test.
Foundation students will study probability, Ratio and proportion, equations & inequalities and
Transformation.
Higher students will cover topics such as Indices, Standard form and surds, Probability, Constructions, loci and congruence, circle theorems.
There will be some early entries in November.
At the end of each topic all students are assessed on a 60 minute end of chapter test.
1st week in December students will sit for a mock Exam.
SPRING TERM 1
5th January-18th February
SPRING TERM 2
28th February-8th April
Foundation students will study Properties of 3D shapes, Constructions and loci under shapes.
They learn index notation, trial and error method, Substitution into expressions and formulae under Algebra.
Higher students will cover topics such as further factorising, simplifying, rational expressions under algebra.
They will study how to estimate mean of group data of statistics.
At end of each topic students are assessed on a 60 minute end of chapter test.
Foundation students will learn how to analysis and draw scatter graphs under data, Pythagoras theorem and 3-D coordinates
Higher students will study topics such as Pythagoras theorem in 3-D and further trigonometry, solving quadratic
simultaneous equations under Algebra, Similarity, similar shapes and
Vectors.
At end of each topic students are assessed on a 60 minute end of chapter test.
Students will sit for a second mock Exam.
SUMMER TERM 1
25th April-27th May
SUMMER TERM 2
6th June-22nd July
Students in foundation groups will start revision on ALEBRA, DATA, NUMBER AND GEOMERY
Most students will practise exam questions and do past papers.
Higher students will study transformation of functions. Once they complete this topic higher students will start their revision on number, shapes, data and algebra and answer exam questions using past exam papers. | 677.169 | 1 |
In this college level Calculus learning exercise, students use the ratio test to determine if a series converges or diverges. The one page learning exercise contains six problems. Solutions are not provided.
Students analyze geometric series in detail. They determine convergence and sum of geometric series, identify a series that satisfies the alternating series test and utilize a graphing handheld to approximate the sum of a series.
In this calculus learning exercise, students find the limit using the Limit Comparison Test and solve problems with series based on the p-series. They tell whether an equation will converge or diverge. There are 7 problems.
In this Calculus worksheet, students assess their understanding of various topics, including the derivatives of trigonometric functions, evaluating integrals, sigma notation, and convergent and divergent series. The one page interactive worksheet contains fifty-two problems. Answers are not provided.
Students investigate sequences and series numerically, graphically, and symbolically. In this sequences and series lesson, students use their Ti-89 to determine if a series is convergent. Students | 677.169 | 1 |
More About
This Textbook
Overview
This is an introduction to Galois Theory along the lines of Galois's Memoir on the Conditions for Solvability of Equations by Radicals. It puts Galois's ideas into historical perspective by tracing their antecedents in the works of Gauss, Lagrange, Newton, and even the ancient Babylonians. It also explains the modern formulation of the theory. It includes many exercises, with their answers, and an English translation of Galois's memoir | 677.169 | 1 |
Numerical analysis, algorithms and computation Murphy J., Ridout D., McShane B., Halsted Press, New York, NY, 1988.Type: Book (9789780470212141) Numerical analysis and computation are widely recognized as important parts of the mathematical modeling of real systems. Given the enormous development and widespread use of computer technology, a new introductory textbook on the numerical... | 677.169 | 1 |
...
Show More time. Furthermore, math teachers and math textbooks simply try to cover too much material, the bulk of which, has no impact on a student's successful completion of math up through calculus in high school. Second, Math Is Easy, So Easy, tries to provide clarity of instruction for a few problems which cover the important aspects of the essential topics. Contrary to most math teacher instruction, it is more important and beneficial to know a few key problems well, than to try to cover many problems only superficially. If you are the parent of a student who is struggling in math, you know how frustrating it can be to get to the bottom of what your student really needs to know to survive and persist in math up through calculus in high school. You also know how important it is that your student stay in math as long as possible in high school, so that they are better prepared to enter and succeed in college. You also, no doubt, know how seemingly unreasonable your struggling student's math teacher can be in terms of communicating with you and your student. As a math teacher for many years now, Max wrote this book to help you and your struggling math student survive math with as few, "I hate math," outbursts as possible. Lastly, Max has personally witnessed many students who struggle in math in high school who then go on to mature into great engineers and scientists. This book will help your student to stay in math longer and be more successful. There is a separate book for each of six math classes: 7th Grade Math, Algebra I, Geometry I, Algebra II, Math Analysis and Calculus. There is a single "Combo" book with all six books in one. Make sure you get the right book for your needs. Nathaniel Max Rock, an engineer by training, has taught math in middle school and high school including math classes: 7th Grade Math, Algebra I, Geometry I, Algebra II, Math Analysis and AP Calculus. Max has been documenting his math curricula since 2002 in various forms, some of which can be found on MathForEveryone.com, StandardsDrivenMath.com and MathIsEasySoEasy.com. Max is also an AVID elective teacher and the lead teacher for the Academy of Engineering at his high school | 677.169 | 1 |
Follows Saxon's Algebra 1, 3rd edition text. Includes lectures and visual instruction for each and every Saxon lesson. Arithmetic and evaluation of expressions involving signed numbers, exponents,...
More about Algebra 1 (3rd ed) Dive Into Math CD
A supplemental compact disc that is designed to be used in conjunction with Saxon's Algebra 1/2 - 2nd edition textbook. Includes lectures and visual instruction for each and every Saxon lesson. ...
More about Algebra 1/2 (2nd ed) Dive Into Math CD
Follows Saxon's 3rd edition textbook. Includes lectures and visual instruction for each each and every Saxon lesson. Fractions, decimals, signed numbers and their arithmetic operations, translating...
More about Algebra 1/2 (3rd ed) Dive Into Math CD
For use with Saxon Algebra 2, 2nd edition textbook. Includes lectures and visual instruction for each and every Saxon lesson. Graphical solutions of simultaneous equations; scientific notation;...
More about Algebra 2 (2nd ed) Dive Into Math CD
The new Online Algebra 1 Placement Test (APT) covers nineteen key concepts essential for success in Algebra 1. It is a quick, easy, and inexpensive way to see if your...
More about Algebra Placement Test
A supplemental CD that may be used in conjunction with the Saxon Calculus, (1st edition) text. Includes lectures and visual instruction for each and every Saxon lesson. Covers all material...
More about Calculus (1st ed) Dive Into Math CD
A collection of word problems with Catholic themes written by the Seton Staff and designed for students using level D of MCP Mathematics. More than thirty separate lessons with 10 Catholic...
More about Catholic Word Problems Level D
A collection of word problems with Catholic themes written by the Seton Staff and designed for students using level E of MCP Mathematics. 15 separate lessons with 10 Catholic word problems...
More about Catholic Word Problems Level E
Counting with Numbers introduces the child to the practice of neatness in all work, carefulness in the use of books, and following directions. It continues the presentation of recognition of shapes...
More about Counting with Numbers
E-Z Grader is a hand-held manual computer. It helps make your homeschooling just a little bit easier. It saves time and effort otherwise needed to compute grades. Gives you freedom to use any number...
More about E-Z Grader
NEW from Dive Into Math, this Interactive CD is for use with the new Saxon Geometry 1st edition textbook. The DIVE CD-ROM teaches each of the 120 lessons and 12 Investigations, plus...
More about Geometry (1st ed) DIVE Into Math CD | 677.169 | 1 |
The best selling 'Algorithmics' presents the most important, concepts, methods and results that are fundamental to the science of computing. It starts by introducing the basic ideas of algorithms, including their structures and methods of data manipulation. It then goes on to demonstrate...
For departments of computer science offering Sophomore through Junior-level courses in Algorithms or Design and Analysis of Algorithms. This is an introductory-level algorithm text. It includes worked-out examples and detailed proofs. Presents Algorithms by type rather than application.
Designed for use in a variety of courses including Information Visualization, Human—Computer Interaction, Graph Algorithms, Computational Geometry, and Graph Drawing. This book describes fundamental algorithmic techniques for constructing drawings of graphs. Suitable as either a textbook ... | 677.169 | 1 |
Based on your search results and the class
discussion which followed the list of on-line tutorial sites for Precalculus has
been pared down to ten (10) sites.
Your job is to evaluate the 10 sites
in order to determine the 2 best sites that you would recommend for on-line
tutorial sites that could be used by Math 105 and Math 120 students and placed
on the Math Department Web page.
You will report your choices as
follows:
A. The address or URL of each on-line site and the name of the
site. (2 points)
Site 1:
Site 2:
B. Describe the strengths of each site and the
weaknesses or drawbacks of each site. Use complete English sentences with proper
grammar and spelling.
Site 1: Strengths: (2 points)
Site 1: Weaknesses: (2 points)
Site 2: Strenghts: (2 points)
Site 2: Weaknesses: (2 points)
C. Prepare a POSTER that would advertise
your number one choice. The poster should include the name of the site, the URL,
emphasize the strengths you identified above and alert the consumer to some of
the weaknesses of the site. You can be as creative as you want. Be sure you
write down the name of the group members on the back of the poster. (15 points) | 677.169 | 1 |
Competency in College Mathematics, 5th Edition, guarantees coverage of the concepts and skills traditionally expected of a liberal arts student. More than 4,000 exercises are presented with answers, along with numerous solved problems, examples and exercises that allow continuous review. Competency in College Mathematics also features the most complete presentation on logic found in any liberal arts mathematics text. A complete testing battery consisting of multiple forms of each chapter test is included upon adoption.
Competency in College Mathematics thoroughly prepares students for the College-Level Academic Skills Test (CLAST) administered by the state of Florida at the completion of the college sophomore year. The book has been revised to reflect the latest CLAST requirements. The Appendix includes an 130-questions sample exam with explained answers. | 677.169 | 1 |
Goals
The Mathematics department develops critical thinking, quantitative literacy, and problem-solving skills in its students through an integrated mathematics curriculum that emphasizes student-centered teaching methodologies, real world applications, and the appropriate use of technology.
The Mathematics Department will...
...create an active learning environment for all its students through the prevalent use of student-centered discovery exercises and cooperative learning strategies.
... foster a deep understanding of mathematical concepts at each grade level by using computers and graphing calculators to solve problems, analyze data, explore patterns, and communicate results.
...promote quantitative literacy and problem-solving skills by frequently using applications from the natural and social sciences, and requiring at least one research project at each grade level.
...establish objectives, select supportive materials, and design assessment instruments in each math course based on a constructivist-learning model, where students build deep understanding through explanation, interpretation, experimentation, application, and reflection.
...publish an integrated model for the mathematics curriculum that identifies key threads and topics that are developed and extended in a progressive flow through consecutive courses. ...employ a wide variety of methods including tests, homework, class work, research reports and oral presentations to assess student levels of critical thinking, quantitative literacy, and problem-solving skills.
...maintain fair and equitable grading practices by aligning assessment techniques with teaching methodologies, and by clearly specifying how students will be assessed in each mathematics course.
...offer a variety of mathematics courses so that students at every ability level have the opportunity to continue and succeed in their mathematics education.
...encourage its faculty members to attend mathematics conferences, enroll in educational courses, and pursue other professional development opportunities that directly support the department's mission and goals.
...strive continuously to make improvements in its curriculum and teaching methods, and annually assess the effectiveness and suitability of all its programs in achieving institutional and departmental goals. | 677.169 | 1 |
Career Opportunities
A degree in mathematics can open the door to a wide range of career opportunities. While most mathematics students seek careers that require them to directly apply their knowledge of mathematics on a daily basis, many others find success in careers that make use of the general problem-solving and logic skills acquired during the study of mathematics.
An excellent resource for mathematical careers is Andrew Serrett's 101 Careers in Mathematics, published by the Mathematical Association of America. This book contains interviews with over one hundred people who have studied mathematics and are now working in a wide variety of areas, including:
Well-known companies such as IBM, FedEx, and L.L. Bean
Government agencies such as the Bureau of the Census and NASA
Legal and medical professions
The field of education at elementary, secondary, and university levels | 677.169 | 1 |
233- Knowing the Vocabulary: A Key to Understanding in College Algebra
Thursday, April 14, 2011: 2:00 PM-3:00 PM
143 (Convention Center)
Lead Speaker:
Susan Gay
Co-Speaker:
Ingrid Peterson
We will present students' work that provides insight into college algebra students' understandings about important concepts such as function, equation and domain. Our data on instructors' and students' opinions about the role of vocabulary will be shared. Then, we'll lead a discussion about implications for high school and college classrooms. | 677.169 | 1 |
The
goal of this course is to
prepare the students to apply quantitative reasoning in work-setting
decisions.† The 501 course takes a
hands-on approach by using real-life examples to illustrate the use of
quantitative tools from algebra, probability and descriptive statistics in
solving concrete problems. This course helps the students to master the
quantitative tools used routinely in the quantitative methods courses: UST601,
UST602, UST803 and in research.† This
course includes computer sessions where the students will be trained to use the
state of the art software MathCAD to solve problems and visualize data.
TENTATIVE
SCHEDULE
®TU, 7/3,
class organization: syllabus, assessment quiz, algebra
®TH, 7/5,
algebra, Hw.1 due, Computer
Lab#1
®TU, 7/10,algebra, Hw.2 due, Computer Lab#2
®TH, 7/12,algebra, Hw.3 due, Computer Lab#3
®TU, 7/17,algebra, Computer Lab #4
®TH, 7/19,algebra, Hw.4 due, Computer Lab #5
®TU, 7/24,Exam I
®TH, 7/26,probability, Computer Lab #6
last day to drop
®TU, 7/31,probability, Hw.5 due, Computer Lab #7
®TH, 8/2, descriptive statistics, Hw.6
due, Computer Lab #8
®TU, 8/7,
descriptive statistics, Hw.7 due,
Computer worksheets: disk and printouts due
®TH, 8/9, Exam
II
The course includes eight
computer sessions.† The grading of the
computer work is based on the number of projects successfully completed and on
the attendance record. You should save each MathCAD worksheet.† A printout of each worksheet will be
collected on Tuesday 8/7/2001.
Each
student needs a calculator.† No textbook
is required.† There are several books in
the library that you may find helpful in learning the material.† They are located in the Rhodes Tower, on the
4íth floor under QA154.† I will
distribute periodically handouts.
Late
homework will not be accepted but in cases of proven reason.† Exam attendance is obligatory.† Makeup exam will be given only in cases of
proven emergency.† Attendance of classes
is needed for the proper understanding of the material.† Class attendance requirements are listed in
the CSU Bulletin. | 677.169 | 1 |
I am going to do MT1002 as a part of my economics degree in September, and I was thinking I should prepare myself by starting to read the course material. But the reading list for the course states 7 different books, and I really don't want do buy them all. Can anyone tell me which book of the following that gives the best overview over the MT1002 course?
Personally I wouldn't buy any textbooks until the lecturers say which ones are the most important, and there will be some in the library anyway. From what I've heard from friends, there was a lot of revision at the beginning of the module (although I suppose it depends on your background in maths) so you should have time to brush up in September before you get to much new stuff. You're really better off asking questions like this in the St Andrews forum btw - someone else there might be able to help you a bit more. | 677.169 | 1 |
I am happy to see that others besides myself have tried to use GAP as
a pedagogical aid in teaching abstract algebra. I am currently teaching
an undergraduate course, using Gallian's book, which I find to be an
excellent text. My students are primarily computer programming majors,
who take abstract algebra because they have to. Thus, one would think
that my class is an ideal laboratory for introducing GAP to students.
However, I can only report limited success. Perhaps some of you in the
forum can give me some suggestions.
I am reluctant to make assignments involving GAP, because I am fairly
new to it myself. I would not know how to evaluate the results. Hence,
the projects I suggest in class are "extra credit". I find the students'
intellectual curiousity is insufficient to cause them to play with GAP
on their own. A manual should include a section telling us mathematicians
how to evaluate computer homework.
I think the GAP manual is pretty intimidating to undergraduates. My students
are struggling with concepts like "isomorphism" and "coset". Even at this
level, they could benefit from some of GAP's capabilities, if they just
ignore all the stuff about character tables, representation theory, etc.
There is a much more user-friendly and simple program called "An Introduction
to Groups / A Computer Illustrated Text" (comes with a disc) by D. Asche,
available from IOP Publihsing for about $40. It does calculations in S_4,
mainly. Even with this, you have to wait until Chapter 5 (in Gallian's text)
before the students can use it. In my class, this is more than halfway through
the first semester. I might consider doing Chapter 5 sooner just so I can
use this software. Still, it seems that programmers ought to be more
interested in GAP. There is a saying, "You can lead a student to a computer,
but you can't make him think." Can we? At the undergraduate level? And with
non-math majors? After I tackle this, I will work on making them like it! | 677.169 | 1 |
Search Course Communities:
Course Communities
Lesson 14: Quadratic Formula
Course Topic(s):
Developmental Math | Quadratics
Completing the square is applied to the general quadratic to derive the quadratic formula. Before an area application example is given, there is a quick review of the four methods that have been presented for solving quadratic equations. Complex numbers are introduced before the discriminant is presented. | 677.169 | 1 |
PEX Quantitative Literacy
GenEd Quantitative Literacy courses present mathematical thinking as a tool for solving everyday problems, and as a way of understanding how to represent aspects of a complexworld. They are designed to prepare students as citizens and voters to have the ability to think critically about quantitative statements, to recognize when they are misleading or false, and to appreciate how they relate to significant social or political issues. While computation may be part a QL course, the primary focus is not computational skills.
Quantitative Literacy courses are intended to teach students how to:
Understand quantitative models that describe real world phenomena and recognize limitations of those models;
Perform simple mathematical computations associated with a quantitative model and make conclusions based on the results;
Recognize, use, and appreciate mathematical thinking for solving problems that are part of everyday life;
Understand the various sources of uncertainty and error in empirical data;
Retrieve, organize, and analyze data associated with a quantitative model; and
Communicate logical arguments and their conclusions.
Courses | 677.169 | 1 |
co730
Our Price
Rs.600
Discount
Rs.131 Master Sat Ii Math 1c And 2c 4th Ed (Master The Sa...
With detailed reviews and expert test-taking strategies, this guide helps prepare you for the exam. It includes extensive review of math subjects ranging from algebra and geometry to trigonometry and statistics. Additional resources include, review questions and full-length practice tests at the end of each chapter to reinforce what you have learned.
Feature and benefits include: - Four full-length practice tests - Diagnostic tests to help students identify the areas in which they need improvement - Detailed review of fundamental subject principles, followed by practice questions | 677.169 | 1 |
Prealgebra (cloth) - 6th edition
Summary: Elayn Martin-Gay firmly believes that every student can succeed, and her developmental math textbooks and video resources are motivated by this belief. ''Prealgebra,'' Sixth Edition was written to help students effectively make the transition from arithmetic to algebra. The new edition offers new resources like the Student Organizer and now includes Student Resources in the back of the book to help students on their quest for success. Whole Numbers and Introduction to Algebra; Intege...show morers and Introduction to Solving Equations; Solving Equations and Problem Solving; Fractions and Mixed Numbers; Decimals; Ratio, Proportion, and Triangle Applications; Percent; Graphing and Introduction to Statistics; Geometry and Measurement; Exponents and Polynomials For all readers interested in prealgebra46 +$3.99 s/h
New
Cloud 9 Books FL West Palm Beach, FL
Hardcover New 032164008X | 677.169 | 1 |
Colin Clark has just published a book Math Overboard.
Photograph by: Photo Submitted
, NEWS
And he's tried to capture some of that knowledge and experience in a new book he has just self-published called Math Overboard.
"It's a book to completely review school mathematics from kindergarten to Grade 12 and make sense of it. Making sense of math could be a subtitle," said Clark, adding it might sound like it's a study book for high school students but it's actually for any adult who wants to understand the logic behind mathematics.
Teaching for as long as he has, the 80-year-old math and physics major said he got the idea for the book after seeing students pay visits to his office time and time again with difficulties grasping concepts.
He didn't have easy answers for them as most resources on the subject didn't delve much deeper than memorizing formulas and read like, well, a math book.
"It goes back to the laws of arithmetic — you need to know the laws to do algebra. It is absolutely fundamental to explain why something is true, such as why it doesn't matter what order you add two numbers together, or multiply two numbers together."
Clark believes the book is of particular use to undergrads, his original intended audience when he started writing.
As a first-year calculus professor on many occasions, Clark witnessed a lot of bright students drop out of their programs or switch majors because they simply couldn't overcome the numbers hurdle. He believes some people may not have received good instruction in pre-university years, were absent during important classes, or just never picked it up.
"I had a student some years ago say 'Look, I can't hope to understand math but I need to pass this course. So if I memorize the techniques, will that work?' said Clark.
"I said 'no, that doesn't work at the university level at all,' and it didn't. She failed."
Math Overboard, which was three years in the making, features problems to solve on every page and diagnostic tests to help readers avoid future errors.
It comes in two parts — the first part, selling for $24 (US), was just released and covers up to Grade 10. The other part will be released in about three months, dealing with advanced topics, such as trigonometry and probability.
Clark has written five other books in the past, but were all intended for entirely different audiences with names like Mathematical Bioeconomics, he said.
Retiring in 1994, Clark has been involved in the field of math all his life. He once considered careers in both engineering and physics, but ended up finding a lot of enjoyment working with biologists.
"They had all the field data, and I had all the mathematical skills they needed to analyze that data," said Clark. | 677.169 | 1 |
0
What it is: I'm not generally one to get excited about math especially algebra, geometry, and *uggh* calculus. But, I think if I had access to a tool like GeoGebra I might have enjoyed them (or at least understood them) more. GeoGebra is a free dynamic geometry system that lets students complete constructions with points, vectors, segments, lines, conic sections, and functions and change them dynamically afterward. Equations and coordinates can also be entered directly; this means thatGeoGebra has the ability to deal with variables for numbers, vectors and points, finds derivatives and integrals of functions and offers commands like Root or Extremum. (If you didn't catch that you are not a high school math teacher *wink*). If you are a high school or college math teacher or know someone who is…that description just made you feel a little excited. GeoGebra is a free multi-platform download.
How to integrate GeoGebra into your curriculum: Use GeoGebra to help your students understand complicated or abstract math concepts. This software is amazing for your visual learners…again a reason this should have existed when I was in school! Allow your students to explore math concepts with this software and to practice their learning. You can also use GeoGebra to create dynamic math worksheets for your students. Very cool!
Tips: Make sure to check out the examples section for some great GeoGebra uses. You can also attend free online workshops to learn how to use GeoGebra. For some great ideas and further explanation of GeoGebra check them out on Wikipedia.
Please leave a comment and share how you are using GeoGebra in your classroom. | 677.169 | 1 |
People who don't learn or understand this material probably won't use
it, but people who do may be surprised to find where it is
useful. This applies not just to the content of the course, but to its
association with careful, creative thinking. It will probably be up to
you to find places where you can use this mathematics. But depending
on your career, you may find that things that are now obvious to you
are not known to others; or on the other hand, you may find it taken
for granted that you know this material and much more. But most
likely, you may actually use the subject of this course and the skills
you've gained, without even realizing it.
In reality, the questions and complaints mentioned above are all too
frequently tacit, and it may be that much more difficult to bring
these issues to a point of real discussion. Sometimes these complaints
only show up on teachers' end-of-term evaluations. There are
certainly more useful responses for individual students in individual
situations than those offered here. The key point, however, is for the
teacher to be able to listen to these kinds of questions and implicit
challenges as having serious substance in them, that strike to the
root of the problems of teaching and learning mathematics.
ACKNOWLEDGMENTS
The authors are grateful to the editor for his very useful
suggestions.
BIOGRAPHICAL SKETCHES
Sandra Keith is a professor of mathematics at St. Cloud State
University MN. Just as Einstein allegedly wanted to ride on a ``beam
of light'', she has been interested in getting into the minds of
students to understand how they think! She has worked with
exploratory writing assignments and other interactive teaching
methods. She served as director and edited Proceedings for the
National Conference of Women in Mathematics and Sciences and was
assistant editor of Winning Women (MAA). Her interests
include better public relations for mathematics, improving the
mathematical environment for women and minorities, better advising,
and mathematical networking.
Jan Cimperman is an assistant professor at the same school. Her
interests include mathematics education, particularly, teaching
elementary teachers. She frequently gives workshops on the MCTM
Standards and the use of manipulatives to explore mathematical
concepts at the K-6 level. She is interested in the variety of ways in
which students learn. | 677.169 | 1 |
Mathematics Course Descriptions
Algebra I
Algebra I is the study of mathematical patterns and ideas. It is balanced between learning skills, exploring concepts, and solving problems. Technology is used to gather, interpret, and represent data from real-world situations. Creating and using mathematical models is a theme throughout. Algebra is integrated with geometry, probability, and statistics. Topics covered include equations-linear, quadratic, and exponential-as well as systems of equations and inequalities, functions, and fractals.
Geometry
This course is investigation-driven and activity-based. It covers topics of Euclidean Geometry such as deductive proof, properties of polygons, circles, similar/congruent triangles, parallel lines, area and volume, the Pythagorean Theorem, basic concepts of right triangle trigonometry, and general ideas of transformations. Computer technology and traditional Geometry tools are used in the investigations. Applications of Geometry concepts to various arts areas are incorporated within the course.
Algebra II
Algebra II is primarily the study of functions-linear, exponential, polynomial, and parametric-through the use of data. Introductory trigonometry, statistics, and probability topics are also explored. Students use calculators, computers, and data gathering devices to investigate all topics. Throughout the course students discover the sense behind the mathematics, rather than simply learn steps for solving problems. Small group work, discussion, and the real world interpretation of the mathematics are stressed. Applications to the arts are woven throughout the curriculum.
Advanced Mathematics
This course is designed to serve students who are preparing for Calculus or further work in mathematics. As a pre-calculus course, it offers an analytical, graphical and numerical approach to understanding polynomials, exponential functions, logarithms, and a wide variety of trigonometry topics. Additional topics may include polar graphs, conic sections, matrices, sequences, and series. Real life applications and data interpretation are integral parts of this course of study.
Advanced Placement Statistics
This course introduces the students to the basic concepts of one of the most important fields of mathematics most people ever encounter. Statistics is about data, and data are numbers with a context. Students learn to make statements of facts and inferences and to state a level of confidence in their inferences. They become proficient in accurately communicating statistical concepts, including methods of data collection and valid interpretations of data. The course follows the topics outlined in the Advanced Placement curriculum in preparation for the AP Test in May.
Advanced Placement Calculus
This course covers approximately one and one-half semesters of college calculus. Students completing the course successfully are prepared to take the AP Calculus AB exam. Topics include limits, continuity, differentiability; optimization, related rates, separable differential equations, and slope fields; indefinite integrals, Riemann Sums, definite integrals, the Fundamental Theorem of Calculus, and applications of the definite integral. The course material is explored through class discussions, small group activities and investigations, sample exam questions, and individual study of problems. | 677.169 | 1 |
Peer Review
Ratings
Overall Rating:
This site consists of a small collection of java applets in elementary mathematics that can be used for classroom demonstrations. It mainly provides supplementary material to enhance visual understanding of concepts that are being explained elsewhere. The applet collection is an incomplete translation of the German version at: (Java 1.1) or (Java 1.4).
Learning Goals:
The applets provide interesting visual representations of several common topics, especially in Geometry, Precalculus and Calculus, and allow interactive manipulation.
Two versions are available using Java 1.1 or Java 1.4. The 1.1 versions run successfully on MACs with older operating systems and browsers and on PC?s with older operating systems. The 1.4 version does not.The Java 1.1 version will not beupdated in future. The Java 1.4 version requires a Java machine of thisversion which can be downloaded from
Evaluation and Observation
Content Quality
Rating:
Strengths:
This site contains a collection of java applets in elementary mathematics ? grouped into Arithmetic (1) , Elementary Geometry (8), Stereometry (2), Spherical Geometry (1), Trigonometry (1), Vector Analysis (2), Analysis (2), Complex Numbers (1).The arithmetic applet, for example, features metric unit conversions for length, area and volume at four different levels of difficulty. The geometry examples such as sum of angles of a triangle and angle subtended in a semicircle were particularly visually attractive. The complex number example provided an attractive combination of graphical and algebraic display that covered multiplication and division as well as addition and subtraction
Concerns:
The arithmetic example did not include instructions for how to enter answers and had some difficulty displaying symbols such as powers. The visual geometry examples did not include any calculations or explanations. The algebraic calculations in the complex number example did not include any intermediate steps.
Potential Effectiveness as a Teaching Tool
Rating:
Strengths:
Paired with an explanation, most of these applets can significantly enhance presentations on geometry. Some applets can be used to present proofs in an intuitive, fairly painless manner. The code can be downloaded and used for non-commercial purposes for those who want to experiment and develop further. Different language versions may be useful in some circumstances ? German, Spanish, Italian, Indonesian and Korean are available
Concerns:
The material would need additional teacher explanations to be useful for most students at this level of content. Some applets are not very convincing demonstrations of the concepts involved; this is mainly due to lack of documentation.
Ease of Use for Both Students and Faculty
Rating:
Strengths:
The applets are generally simple and uncluttered with a strong visual effect.
Concerns:
The instructions for using the applets are cursory or non-existent ? for example how to enter answers in the unit conversion applet. There are no background explanations of the mathematics involved so these would be needed to supplement visual understanding.
Other Issues and Comments:
Useful as a supplement to other explanations of the material. Physics and astronomy applets are also available at the same site. | 677.169 | 1 |
Curriculum Design: Pre-requisites/Co-requisites/Exclusions
To introduce the basic concept of a limit, together with the derived concepts of convergent series, continuous functions and differentiation. To present the most important results connected with these concepts.
At the end of the module students should be able to • quote and understand the definition of a limit of a sequence or a function in its various forms; • understand proofs using these definitions, and write simple examples of such proofs; • demonstrate the convergence or divergence of the geometric and harmonic series and other standard series • know and apply the basic tests for convergence of infinite series; • calculate limits of particular functions involving products and ratios of polynomials and power series; • understand the proofs of the intermediate value theorem and the theorem on boundedness of continuous functions, and apply these theorems; | 677.169 | 1 |
A Level Further Mathematics
Introduction:
If you choose Further Maths you will be studying for two A-levels in Maths. This could mean that you spend half your time in College doing Maths. You have to really enjoy the subject. It is highly recommended if are considering studying a degree in Maths, Physics, Engineering or Economics at one of the best universities. Although you will get a whole A-level in a year it is not recommended for students who just want to get their Maths out of the way and do not intend to continue to the second year. It may be possible to pick up an AS in Further Maths in the second year which would involved just three of the six extra modules. | 677.169 | 1 |
Mathematics
Page Content
MATH 100. Basic College Mathematics (3; F, S)
Three hours per week. This
course may not be used to satisfy the University's Core mathematics
requirement. Students may not enroll in this course if they have
satisfactorily completed a higher numbered MATH course. An overview of basic
algebraic and geometric skills. This course is designed for students who lack
the needed foundation in college level mathematics. A graphing calculator is
required.
MATH 104. College Algebra (3; F, S) Three hours per week.
Prerequisite: MATH 100. This course may
not be used to satisfy the University's Core mathematics requirement.
Qualitative and quantitative aspects of linear, exponential, rational, and
polynomial functions are explored using a problem solving approach. Basic
modeling techniques, communication, and the use of technology is emphasized. A
graphing calculator is required.
MATH 110. The Mathematics of Motion & Change (3; F, S) Three hours per week. Prerequisite: MATH 104. A study of the mathematics of
growth, motion and change. A review of algebraic, exponential, and trigonometric
functions. This course is designed as a terminal course or to prepare students
for the sequence of calculus courses. A graphing calculator is required.
MATH 112. Modern Applications of Mathematics (3; F, S)
Three hours per week. Prerequisite: MATH 104. Calculus concepts as
applied to real-world problems. Topics include applications of polynomial and
exponential functions and the mathematics of finance. A graphing calculator is
required.
MATH 140. Calculus I (4; F, S) Four hours per week.
Prerequisite: A "C" or better in MATH 110. Rates of change, polynomial and
exponential functions, models of growth. Differential calculus and its
applications. Simple differential equations and initial value problems. A
graphing calculator is required.
MATH 141. Calculus II (4; F, S) Four hours per week.
Prerequisite: A "C" or better in MATH 140. The definite integral, the
Fundamental Theorem of Calculus, integral calculus and its applications. An
introduction to series including Taylor series and its convergence. A graphing
calculator is required.
MATH 150. Introduction to Discrete Structures (3; S) Three hours per week. Prerequisite: A "C" or better in one of MATH 110, MATH
112 or MATH 140. An introduction to the mathematics of computing. Problem
solving techniques are stressed along with an algorithmic approach. Topics
include representation of numbers, sets and set operations, functions and
relations, arrays and matrices, Boolean algebra, propositional logic, big O and
directed and undirected graphs.
MATH 199. Special Topics (var. 1-4; AR) May be repeated
for credit when topic changes. Selected topics of student interest and
mathematical significance will be treated.
MATH 206. Statistical Methods in Science (4; S) Four
hours per week. Prerequisite: A "C" or better in MATH 140. Credit cannot be awarded for both MATH 205
and MATH 206. Concepts of probability, distributions of random variables,
estimation, hypothesis testing, regression, ANOVA, design of experiments,
testing of assumptions, scientific sampling and use of statistical software.
Many examples will use real data from scientific research. A graphing calculator
is required.
MATH 220WI. Mathematics & Reasoning (3; S) Three
hours per week. Prerequisite: ENGL 103 and a "C" or better in MATH 141.
Fundamentals of mathematical logic, introduction to set theory, methods of proof
and mathematical writing.
MATH 306. Regression & Analysis of Variance Techniques
(3) Three hours per week. Prerequisites: A "C" or better in MATH
141, and a "C" or better in either MATH 205 or MATH 305. Theory of least
squares, simple linear and multiple regression, regression diagnostics, analysis
of variance, applications of techniques to real data and use of statistical
packages.
MATH 307. College Geometry (3) Three hours per week.
Prerequisite: A "C" or better in MATH 141. A critical study of deductive
reasoning used in Euclid's geometry including the parallel postulate and its
relation to non-Euclidean geometries.
MATH / PHIL 330. Symbolic Logic (3) Three hours per week.
A study of modern formal logic, including both sentential logic and predicate
logic. This course will improve students' abilities to reason effectively.
Includes a review of topics such as proof, validity, and the structure of
deductive reasoning.
MATH 351. Applied Mathematics (3; F) Three hours per
week. Prerequisite: A "C" or better in both MATH 300 and MATH 331. Advanced
calculus and differential equations methods for analyzing problems in the
physical and applied sciences. Calculus topics include potentials, Green's
Theorem, Stokes' Theorem, and the Divergence Theorem. Differential equations
topics include series solutions, special functions, and orthogonal
functions.
MATH 354. Introduction to Partial Differential Equations and Modeling
(3; S) Three hours per week. Prerequisite: A "C" or better in both
MATH 300 and MATH 331. Modeling problems in the physical and applied sciences
with partial differential equations, including the heat, potential, and wave
equations. Solution methods for initial value and boundary value problems
including separation of variables, Fourier analysis, and the method of
characteristics.
MATH 400SI. History of Mathematics (3) Three hours per
week. Prerequisite: A "C" or better in MATH 220WI and junior or senior status.
This course may not be used to satisfy
the University's Core mathematics requirement. A study of the history of
mathematics. Students will complete and present a research paper. Students will
gain experience in professional speaking.
MATH 411. Introduction to Real Analysis (3) Three hours
per week. Prerequisite: A "C" or better in both MATH 220WI and MATH 300.
Foundations of real analysis including sequences and series, limits, continuity,
and differentiability. Emphasis on the rigorous formulation and writing of
proofs.
MATH 412. Introduction to Complex Variables (3) Three
hours per week. Prerequisite: A "C" or better in both MATH 220WI and MATH 300.
Algebra of complex numbers, analytic functions, elementary functions, line and
contour integrals, series, residues, poles and applications.
MATH 423. Algebraic Structures (3) Three hours per week.
Prerequisite: A "C" or better in MATH 220WI. An overview of groups, rings,
fields and integral domains. Applications of abstract algebra.
MATH 440. Special Topics (var. 1-3; AR) Prerequisite: A
"C" or better in MATH 220WI or consent of the instructor. May be repeated for
credit when topic changes. Selected topics of student interest and mathematical
significance will be treated.
MATH 501. Introduction to Analysis (3) Three hours per
week. A study of real numbers and the important theorems of differential and
integral calculus. Proofs are emphasized, and a deeper understanding of calculus
is stressed. Attention is paid to calculus reform and the integrated use of
technology.
MATH 502. Survey of Geometries (3) Three hours per week.
An examination of Euclidean and non-Euclidean geometries. Transformational and
finite geometries.
MATH 503. Probability & Statistics (3) Three hours
per week. Probability theory and its role in decision-making, discrete and
continuous random variables, hypothesis testing, estimation, simple linear
regression, analysis of variance and some nonparametric tests. Attention is paid
to statistics reform and the integrated use of technology.
MATH 504. Special Topics (3; AR) Three hours per week.
May be repeated for credit when topic changes. Course content will vary
depending on needs and interests of students.
MATH 507. Number Theory (3) Three hours per week. An
introduction to classical number theory. Topics include modular arithmetic, the
Chinese Remainder Theorem, primes and primality testing, Diophantine equations,
multiplicative functions and continued fractions.
MATH 510. Seminar in the History of Mathematics (3) Three hours per week. Important episodes, problems and discoveries in
mathematics, with emphasis on the historical and social contexts in which they
occurred.
MATH 515. Combinatorics (3) Three hours per week. A
survey of the essential techniques of combinatorics. Applications motivated by
the fundamental problems of existence, enumeration and optimization.
MATH 520. Linear Algebra (3) Three hours per week.
Applications of concepts in linear algebra to problems in mathematical modeling.
Linear systems, vector spaces and linear transformations. Special attention will
be paid to pedagogical considerations.
MATH 531. Theory of Ordinary Differential Equations (3) Three hours per week. Existence and uniqueness theorems. Qualitative and
analytic study of ordinary differential equations, including a study of first
and second order equations, first order systems and qualitative analysis of
linear and nonlinear systems. Modeling of real world phenomena with ordinary
differential equations.
MATH 600. Thesis Seminar (1-3) One to three hours per
week. Research guidance. May be repeated for credit up to a total of three
semester hours.
MATH 699. Thesis Preparation and Research (1) Master of
Arts in Mathematics students who have not completed their thesis and are not
enrolled in any other graduate course must enroll in MATH 699 each fall and
spring semester until final approval of their thesis. This course is Pass/Fail
and does not count towards any graduate degree. | 677.169 | 1 |
* Links the maths with the chemical applications in integrated, double page presentations, helping the student to appreciate and understand the relevance and importance of the maths.
* Detailed guidance on the mechanics of the mathematical manipulations required, set in the context of chemistry, helps to develop the student's mathematical ability and understanding.
* Practice problems (with answers) at the end of each section include both simple mathematical practice and real chemical examples, to give the student a thorough grounding in the mathematical techniques required.* More detailed problems, again with answers and covering a wider range of chemical themes, at the end of each broad topic give the student the ability and confidence to recognise when particular mathematical manipulations are required, and to apply them when necessary. | 677.169 | 1 |
Jetzt kaufen
Langtext Principles of Linear Algebra with Mathematica(r) uniquely addresses the quickly growing intersection between subject theory and numerical computation. Computer algebra systems such as Mathematica(r) are becoming ever more powerful, useful, user friendly and readily available to the average student and professional, but thre are few books which currently cross this gap between linear algebra and Mathematica(r). This book introduces algebra topics which can only be taught with the help of computer algebra systems, and the authors include all of the commands required to solve complex and computationally challenging linear algebra problems using Mathematica(r). The book begins with an introduction to the commands and programming guidelines for working with Mathematica(r). Next, the authors explore linear systems of equations and matrices, applications of linear systems and matrices, determinants, inverses, and Cramer's rule. Basic linear algebra topics, such as vectors, dot product, cross product, vector projection, are explored as well as the more advanced topics of rotations in space, rolling a circle along a curve, and the TNB Frame. Subsequent chapters feature coverage of linear programming, linear transformations from Rn to Rm, the geometry of linear and affine transformations, and least squares fits and pseudoinverses. Although computational in nature, the material is not presented in a simply theory-proof-problem format. Instead, all topics are explored in a reader-friendly and insightful way. The Mathematica(r) software is fully utilized to highlight the visual nature of the topic, as the book is complete with numerous graphics in two and three dimensions, animations, symbolic manipulations, numerical computations, and programming. Exercises are supplied in most chapters, and a related Web site houses Mathematica(r) code so readers can work through the provided examples.
Aus dem Inhalt Preface.
Conventions and Notations.
1. An Introduction to Mathematica.
1.1 The Very Basics.
1.2 Basic Arithmetic.
1.3 Lists and Matrices.
1.4 Expressions Versus Functions.
1.5 Plotting and Animations.
1.6 Solving Systems of Equations.
1.7 Basic Programming.
2. Linear Systems of Equations and Matrices.
2.1 Linear Systems of Equations.
2.2 Augmented Matrix of a Linear System and Row Operations.
2.3 Some Matrix Arithmetic.
3. Gauss-Jordan Elimination and Reduced Row Echelon Form.
3.1 Gauss-Jordan Elimination and rref.
3.2 Elementary Matrices.
3.3 Sensitivity of Solutions to Error in the Linear System.
4. Applications of Linear Systems and Matrices.
4.1 Applications of Linear Systems to Geometry.
4.2 Applications of Linear Systems to Curve Fitting.
4.3 Applications of Linear Systems to Economics.
4.4 Applications of Matrix Multiplication to Geometry.
4.5 An Application of Matrix Multiplication to Economics.
5. Determinants, Inverses, and Cramer' Rule.
5.1 Determinants and Inverses from the Adjoint Formula.
5.2 Determinants by Expanding Along Any Row or Column.
5.3 Determinants Found by Triangularizing Matrices.
5.4 LU Factorization.
5.5 Inverses from rref.
5.6 Cramer's Rule.
6. Basic Linear Algebra Topics.
6.1 Vectors.
6.2 Dot Product.
6.3 Cross Product.
6.4 A Vector Projection.
7. A Few Advanced Linear Algebra Topics.
7.1 Rotations in Space.
7.2 "Rolling" a Circle Along a Curve.
7.3 The TNB Frame.
8. Independence, Basis, and Dimension for Subspaces of Rn.
8.1 Subspaces of Rn.
8.2 Independent and Dependent Sets of Vectors in Rn.
8.3 Basis and Dimension for Subspaces of Rn.
8.4 Vector Projection onto a subspace of Rn.
8.5 The Gram-Schmidt Orthonormalization Process.
9. Linear Maps from Rn to Rm.
9.1 Basics About Linear Maps.
9.2 The Kernel and Image Subspaces of a Linear Map.
9.3 Composites of Two Linear Maps and Inverses.
9.4 Change of Bases for the Matrix Representation of a Linear Map.
10. The Geometry of Linear and Affine Maps.
10.1 The Effect of a Linear Map on Area and Arclength in Two Dimensions.
10.2 The Decomposition of Linear Maps into Rotations, Reflections, and Rescalings in R2.
10.3 The Effect of Linear Maps on Volume, Area, and Arclength in R3.
10.4 Rotations, Reflections, and Rescalings in Three Dimensions.
10.5 Affine Maps.
11. Least-Squares Fits and Pseudoinverses.
11.1 Pseudoinverse to a Nonsquare Matrix and Almost Solving an Overdetermined Linear System.
11.2 Fits and Pseudoinverses.
11.3 Least-Squares Fits and Pseudoinverses.
12. Eigenvalues and Eigenvectors.
12.1 What Are Eigenvalues and Eigenvectors, and Why Do We Need Them?
12.2 Summary of Definitions and Methods for Computing Eigenvalues and Eigenvectors as well as the Exponential of a Matrix.
12.3 Applications of the Diagonalizability of Square Matrices.
12.4 Solving a Square First-Order Linear System if Differential Equations. | 677.169 | 1 |
This
course is a continuation of Analytic Geometry and Calculus II,
extending the skills of differentiation and integration by learning new
techniques and working with partial derivatives and double and triple
integrals. Other major topics include cylindrical and spherical
coordinates, quadric surfaces, vector functions, vector analysis,
Green''s theorem and Stoke''s theorem.
GENERAL EDUCATION APPLICABILITY
CSU GE Area B: Physical and its Life Forms(mark all that apply) = B4 - Mathematics/Quantitative Thinking;
UC Transfer Course:
CSU Transfer Course:
STUDENT LEARNING OUTCOMES Upon completion of the course, the student will be able to
Use the Cartesian, polar, cylindrical, and spherical coordinate systems effectively.
Use scalar and vector products in applications.
Use vector-valued functions to describe motion in space.
Extend the concepts of derivatives, differentials, and integrals to include multiple independent variables.
Solve simple differential equations of the first and second order.
REQUISITES
Prerequisite:
MATH C152A. Partial Differentiation
1. Functions of two or more variables
a. Limits
b. Continuity
c. Geometric interpretation
d. Derivatives
2. Tangent planes and normal lines
3. The directional derivative
4. The gradient
5. The chain rule
6. Linearization and differentials
7. Maximum-Minimum problems
a. Use of derivatives for extreme values
b. Lagrange multipliers
c. Methods of least squares
8. Higher order derivatives
B. Multiple Integrals
1. Functions of two or more variables
a. Plane area
b. Volume
c. Center of mass
d. Moments of inertia
e. Polar coordinates
f. Surface area
2. Triple Integrals
a. Volume
b. Center of mass
c. Moments of inertia
d. Cylindrical coordinates
e. Spherical coordinates
C. Vectors and Parametric Equations
1. Parametric Equations in Kinematics
2. Parametric Equations in Analytic Geometry
3. Vectors in two dimensions
a. The i and j components
b. Vector algebra
c.
Unit and Zero
Vectors
4. Space Coordinates
a. Cartesian
b. Cylindrical
c. Spherical
5. Vectors in Space
6. Scalar Product of Two Vectors
a. Algebraic properties
b. Orthogonal vectors
c. Vector projection
7. Vector Product of Two Vectors
a. Algebraic properties
b. Area
8. Equations of Lines and Planes
9. Product of Three or More Vectors
10. Cylinders
11. Quadric Surfaces
D. Vector Functions and Their Derivatives
1. Derivative of a Vector Function
2. Velocity and Acceleration
3. Tangential Vectors
4. Curvature and Normal Vectors
5. Differentiation of Products of Vectors
6. Polar and Cylindrical Coordinates
E. Multi-Dimensional Vector Analysis
1. Vector fields
2. Surface integrals
3. Line integrals
4. Green's Theorem
5. Stokes' Theorem
METHODS OF INSTRUCTION--Course instructional methods may include but are not limited to
Discussion;
Lecture;
Other Methods: A. lecture and discussion of all course concepts.
B. demonstration of developing proofs and solving application problems.
C. reading textbooks and journals to see presentations different than those of the instructor.
D. assignments and quizzes
E. the use of computational and other types of mathematical software
OUT OF CLASS ASSIGNMENTS: Out of class assignments may include but are not limited to
A. Reading assignments.
B. Bi-weekly homework assignments.
METHODS OF EVALUATION: Assessment of student performance may include but is not limited to
A. tests on course content, to include solving equations as well as demonstration of specific skills B. quizzes (in-class and take-home) to include solving equations as well as demonstration of specific skills C. group work to analyze and solve application problems
TEXTS, READINGS, AND MATERIALS: Instructional materials may include but are not limited to | 677.169 | 1 |
geometr... read more
Regular Polytopes by H. S. M. Coxeter Foremost book available on polytopes, incorporating ancient Greek and most modern work. Discusses polygons, polyhedrons, and multi-dimensional polytopes. Definitions of symbols. Includes 8 tables plus many diagrams and examples. 1963 edition.
Shape Theory: Categorical Methods of Approximation by J. M. Cordier, T. Porter This in-depth treatment uses shape theory as a "case study" to illustrate situations common to many areas of mathematics, including the use of archetypal models as a basis for systems of approximations. 1989 editionProjective Geometry by T. Ewan Faulkner Highlighted by numerous examples, this book explores methods of the projective geometry of the plane. Examines the conic, the general equation of the 2nd degree, and the relationship between Euclidean and projective geometry. 1960 edition.
Non-Riemannian Geometry by Luther Pfahler Eisenhart This concise text by a prominent mathematician deals chiefly with manifolds dominated by the geometry of paths. Topics include asymmetric and symmetric connections, the projective geometry of paths, and the geometry of sub-spaces. 1927 edition.
The Elements of Non-Euclidean Geometry by D. M.Y. Sommerville Renowned for its lucid yet meticulous exposition, this classic allows students to follow the development of non-Euclidean geometry from a fundamental analysis of the concept of parallelism to more advanced topics. 1914 edition. Includes 133 figuresProduct Description:
geometry plays in a wide range of mathematical applications.
Bonus Editorial Feature:
Harold In the Author's Own Words: "I'm a Platonist — a follower of Plato — who believes that one didn't invent these sorts of things, that one discovers them. In a sense, all these mathematical facts are right there waiting to be discovered."
"In our times, geometers are still exploring those new Wonderlands, partly for the sake of their applications to cosmology and other branches of science, but much more for the sheer joy of passing through the looking glass into a land where the familiar lines, planes, triangles, circles, and spheres are seen to behave in strange but precisely determined ways."
"Geometry is perhaps the most elementary of the sciences that enable man, by purely intellectual processes, to make predictions (based on observation) about the physical world. The power of geometry, in the sense of accuracy and utility of these deductions, is impressive, and has been a powerful motivation for the study of logic in geometry."
"Let us revisit Euclid. Let us discover for ourselves a few of the newer results. Perhaps we may be able to recapture some of the wonder and awe that our first contact with geometry aroused." — H. S. M. Coxeter Harold | 677.169 | 1 |
All Faculty, Staff and Students can download and install Mathematica from Wolfram. All you need is to set up an account on Wolfram's webpage with your SSU email account. Here are the directions on how to do this.
Students who are interested in sports can participate in Intramurals or Sports Clubs. Intramural sports offer students the opportunity to create their own teams and compete against other SSU players. You can visit our Sports Clubs section here.
Interested in an amazing student driven environment where community, human awareness & diversity, leadership, service, and FUN is paramount? MOSAIC, an acronym for "Making Our Space An Inclusive Community", is a living and learning community made with you in mind.
Syntactics: Relations among signs in formal structures
Pragmatics: Relation between signs and the effects they have on the people who use them
There is something unsatisfying and lacking, however, in the concept of the body, which undermines the very effort to ground (mathematical) knowledge differently than in the private cogitations of the isolated mind. The purpose of this paper is to argue for a more radical approach to the conceptualization of mathematical knowledge that is grounded in dialectical materialist psychology (as developed by Lev Vygotsky), materialist phenomenology (as developed by Maine de Biran and Michel Henry),
He presents a way of understanding knowing and learning in mathematics that differs from other current approaches, using case studies to demonstrate contradictions and incongruences of other theories–Immanuel Kant, Jean Piaget, and more recent forms of (radical, social) constructivism, embodiment theories, and enactivism–and to show how material phenomenology fused with phenomenological sociology provides answers to the problems that these other paradigms do not answer. | 677.169 | 1 |
Catalog Description: This course is intended to be a one-semester survey of Calculus topics specifically for Biology majors. Topics include limits, derivatives, integration, and their applications, particularly to problems related to the life sciences. The emphasis throughout is more on practical applications and less on theory. Pre-requisite: grade of C in Math 180, or suitable placement score. This course qualifies as a General Education course, G9.
Text:Calculus for the Life Sciences, by Bittinger, Brand, and Quintanilla (Pearson-Addison-Wesley, 2006).
Note: I have chosen a text that includes a nice collection of problems that are oriented toward applications of calculus to the life sciences. So this course really has two characteristics: (1) it is something of a "Calculus lite", a little less emphasis on theory and proof that the traditional course, and (2) it has a particular emphasis on applications in the life sciences.
General Education Core Skill Objectives
1. Thinking Skills: Students engage in the process of inquiry and problem solving, which involves both critical and creative thinking.
(a) The student understands the "big problems" in the development of differential calculus, the tangent problem and the area under the curve problem.
(b) The student understands the mathematical concept of Limit.
(c) The student explores differentiation and works with differentiation formulas for a variety of functions, including exponential and logarithmic functions, and the applications of these methods, especially to problems from the realm of life sciences
(d) The student explores integration and a variety of integration techniques, and applications of these techniques to a variety of problems, especially those related to the life sciences.
2. Communication Skills: Students communicate effectively orally and in writing in an appropriate manner both personally and professionally.
(a) The student does group work (labs and practice exams) throughout the course, involving both written and oral communication.
(b) The student uses technology - graphing calculators and DERIVE in the computer labs - to solve problems and to be able to communicate solutions and explore options.
(c) The student will use the language of mathematics accurately and appropriately.
(d) The student will present mathematical content and argument in written form.
3. Life Values: Students analyze, evaluate, and respond to ethical issues from informed personal, professional, and social value systems.
(a) The student develops an appreciation for the intellectual honesty of deductive reasoning.
(b) The student understands the need to do one's own work, to honestly challenge oneself to master the material.
4. Cultural Skills: Students understand their own and other cultural traditions and demonstrate a respect for the diversity of the human experience.
(a) The student develops an appreciation of the history of calculus and the role it has played in mathematics and in other disciplines.
(b) The student learns to use the language of mathematics - symbolic notation - correctly and appropriately.
5. Technology: (a) The student will demonstrate the basic ability to perform computational and algebraic procedures using a calculator or computer.
(b) The student will demonstrate the ability to efficiently and accurately graph functions using a calculator or computer.
(c) The student will demonstrate the knowledge of the limitations of technological tools.
(d) The student will demonstrate the ability to work effectively with a CAS, such as DERIVE, to do a variety of mathematical work.
Required Course Work Your basic work in this course is to learn the material and develop your problem-solving kills so that you can apply the concepts and methods of the calculus to problems you will encounter along the way. It is important that you attend class regularly and especially that you do the homework. The homework may seem "hidden" to you since it will not be graded, but it precisely in that outside-of-class practice that you learn the material. There is a rule of thumb that says college courses require roughly 2 hours outside of class for every hour in class and I think this is not at all an overstatement. It takes discipline to leave class on a Wednesday or Thursday, knowing that you have class again the very next day, and still find a couple hours to practice your calculus, but it is exactly that sort of discipline that is required for success in the course. In short: DO YOUR HOMEWORK!
Your grade will be based on two primary factors: (1) Exams; and (2) Group work. There will be a quiz following the review chapter, an exam following each of chapters 2 , 3, and 4, and then a final cumulative exam at the end of the course, following chapter 5. These exams will be done in class subject to the 50 minute time limit (except for the final, of course), but you can use a set of notes you prepare for the exam and you may use a graphing calculator as well. I see exams as a way to see if you as an individual can actually do what the course asks of you.
Secondly, there will be two types of groupassignments: (a) Labs; and (b) practice exams. Every so often, roughly once a chapter, I will take a day and have you work in small groups of 2 or 3 students on a set of problems. I will refer to these problem sets as "labs", in the sense that you will be exploring the concepts and trying to apply them to the problems. I like the idea of asking you to occasionally work in groups – I think the opportunity to discuss the mathematical ideas is extremely valuable. After all, mathematics is a language as well as a way of thinking about things.
I will also have you take a "practice exam" during the class period just prior to each individual exam. These practice exams will be done in groups like the labs, and must be turned in by the end of the period, as do the labs. This is an attempt on my part to give you an idea of what to expect on the exam.
I am also going to try a sort of "oral exam" this semester – let's call it an "interview". After the third exam I will meet individually with each of you and ask a few questions to test your understanding of the material we have been covering.
It is also worth mentioning here that because one of our goals is to develop some skills with technological tools, I will be asking you to use DERIVE on a few problems along the way. DERIVE is a computer program created by Texas Instrument, the calculator people, to perform a very wide variety of mathematical procedures. It is a CSA (Computer Algebra System), which means it can perform algebraic manipulations like solving equations or factoring polynomials, as well as graphing functions in 2 or 3 dimensions. It can also perform calculus procedures and we will have a chance to see some of this power.
Instructional Methods My general "lecture" style is more of a give-and-take discussion than simply a rote presentation of material. I like to try to see that the class is following what I am doing and so I want feedback along the way. I often will ask a leading question to see that you are ready for what is about to come.
One specific strategy I will use is called Think-Pair-Share. In this case I will ask a question and have each of you think about it for a bit, then pair up with a "neighbor" and finally share the answer with the rest of the class. Not every pair will share with the class each time, but I hope as we do this on a regular basis most of you will have an opportunity to speak.
I have already mentioned the "Labs" we will do on occasion. I will randomly assign you to a group and give each of you rotating roles within the process; I'll explain the details when we do that first lab on Wednesday, 7 September. I will try to keep the labs reasonably short so that the group can turn in the work at the end of the period.
Assessment Strategies The main outcomes I want to see in each of you can basically be listed briefly as: (1) thinking, or problem solving; (2) communicating your mathematical ideas and solutions; and (3) using technology to do some of the work. I will be assessing your ability to do these things throughout the course by grading the labs, the technology assignments, and the exams. I will also be noting your participation in the class room, both in your group work and in general class discussion.
Grading System I expect, in sum, that we will have the following possible points during the semester:
I will then assign letter grades as follows: 90% of possible points for "A", 80% for a "B", 70% for a "C", and 60% for a "D". By the way, I am aware that the biology program requires a grade of at least a "C" in its support courses, so you needn't tell me that if the going gets "close" later on in the course. I wish this wasn't the case, since it is sometimes stressful on me as well as on you, and I don't like "losing" the possibility of giving a "D" grade – sometimes people pass a course but "just barely".
One final note about grading: I will "guarantee" the grade you earn on your final exam! What I mean by this is, regardless of the grade you have earned based on the percentage of total points, if you score a higher grade on the final I will accept this demonstration that you have learned the material at this higher level and will give that grade. Please don't depend on this as a means of getting through the course – most people do not show better performance on the final exam, but it does give you one last chance to improve things.
Attendance Policy I think that regular attendance is of paramount importance in any course, but perhaps a mathematics course more than most. There may be the occasional exception, a student who is so mathematically talented that she or he can get the material by just reading the text and doing some problems, but for most students it is important to be in class every day. To encourage this I am going to include attendance in the grading scheme in a simple way: 1 point for each day you are there. I'm not going to engage in deciding whether absence is "excused" – too many subtleties and degrees there - just a point a day when you are there.
Disability Statement I want to include taking exams in the learning center under this category; you will need a written request from Wayne Wojciechowski before I will allow you to take exams there. | 677.169 | 1 |
Mathematics 2205/3205 represents the new level II and III Honors
Mathematics courses being taught for the first time this year. Mathematics
2205 is taught during Term I and Mathematics 3205 is taught during Term
II. Taught at a very fast pace, these courses are designed for students
with a strong aptitude for mathematics and those who plan on attending
university following high school. Both courses rely heavily on the use of
the TI - 83 Plus Graphic Calculator and are taught utilizing a great deal of
technology throughout. The texts for the courses are Mathematics Modeling
II and Mathematical Modeling III. The final exam in Mathematics 3205 will
be a public exam delivered by the Provincial Department of Education. | 677.169 | 1 |
Statistics reveal that more than two-thirds of students fear mathematics as a subject. Needless to mention, most students loathe mathematics assignments. However, it is good to know that mathematics is one of the most interesting subjects, and if students take the suggestions and tips mentioned in this article, then they are sure to be able to present a decent assignment | 677.169 | 1 |
Subsets and Splits
No saved queries yet
Save your SQL queries to embed, download, and access them later. Queries will appear here once saved.